WO2022009761A1

WO2022009761A1 - Optimization problem solution and optimization problem solution apparatus

Info

Publication number: WO2022009761A1
Application number: PCT/JP2021/024899
Authority: WO
Inventors: 直紀井手
Original assignee: ソニーグループ株式会社
Priority date: 2020-07-10
Filing date: 2021-07-01
Publication date: 2022-01-13
Also published as: US20230342416A1; EP4181034A1; CN115769237A; EP4181034A4; JPWO2022009761A1

Abstract

In an optimization problem solution according to the present disclosure, first data from a sensor is inputted, an objective function for sparse modeling on the first data is generated, a coefficient matrix for a variable to be optimized in the objective function is generated, the coefficient matrix is transmitted to a first Ising machine for performing a combinatorial optimization calculation, and the optimal solution of sparse modeling is generated on the basis of second data received from the first Ising machine.

Description

Optimization problem solving method and optimization problem solving device

This disclosure relates to an optimization problem solving method and an optimization problem solving device.

Conventionally, as hardware that imitates the Ising model (hereinafter also referred to as "Ising machine") for solving combinatorial optimization problems (hereinafter also referred to as "optimization problem"), quantum annealing machines and CMOS that use qubits for Ising spins. (Complementary metal-oxide-semiconductor) Quantum-conceived computers that use gates, digital circuits, etc. are known. In such an Ising machine, in order to solve the combinatorial optimization problem, the process of converting the evaluation function has been performed. In addition, sparse modeling has been used in the processing of the conversion of the evaluation function so as to satisfy the restrictions in the hardware (for example, Patent Document 1).

Japanese Unexamined Patent Publication No. 2019-046038

According to the prior art, sparse modeling was used to perform the conversion process of the evaluation function to match the hardware constraints on the rising machine prepared to solve the optimization problem.

However, the method of realizing sparse modeling used here uses a conventional algorithm. Originally, sparse modeling is widely known as an optimization problem for modeling data from sensors and the like with as few explanatory variables as possible. However, it is also known that the method that is expected to have the highest accuracy in the method of realizing sparse modeling is a combinatorial optimization problem that causes a combinatorial explosion and is not practical. On the other hand, Ising machines have recently been in the limelight as a method of reducing calculation time for combinatorial optimization that causes a combinatorial explosion.

Therefore, this disclosure proposes an optimization problem solving method and an optimization problem solving device that can appropriately generate a solution of an optimization problem in sparse modeling for data from a sensor using a sparse machine.

In order to solve the above problems, one form of the optimization problem solving method according to the present disclosure inputs the first data from the sensor, generates an objective function for sparse modeling the first data, and then generates an objective function. A coefficient matrix related to a variable to be optimized in the objective function is generated, the coefficient matrix is transmitted to a first Ising machine that performs combinatorial optimization calculation, and based on the second data received from the first Ising machine. Generate the optimal solution for sparse modeling.

It is a figure which shows the configuration example of the optimization problem solution system of this disclosure. It is a block diagram of 1st Example. It is a flowchart which shows the processing procedure about a user interface. It is a figure which shows an example of a user interface. It is a figure which shows the Ising model schematically. It is a figure which shows the tunnel effect schematically. It is a conceptual diagram showing sparse modeling. It is a conceptual diagram which shows the determinant about a binary conversion. It is a flowchart which shows the procedure of the process of L0 sparse modeling. It is a figure which shows an example of the program of L0 sparse modeling. It is a figure which shows an example of the program which generates the objective function of the Ising model. It is a figure which shows an example of the program which generates the solution of L0 sparse modeling. It is a figure which shows the configuration example of the optimization problem solving apparatus of this disclosure. It is a figure which shows the structural example of the Ising machine of this disclosure. It is a flowchart which shows the procedure of the compressed sensing optimization processing. It is a figure which shows an example of the program of the compressed sensing optimization processing. It is a figure which shows an example of the program which generates the objective function of the Ising model. It is a figure which shows an example of the program which obtains a measurement control matrix. It is a figure which shows the 2nd Example which applied the optimization problem solution system. It is a figure which shows the 4th Example which applied the optimization problem solution system. It is a figure which shows the 4th Example which applied the optimization problem solution system. It is a figure which shows the 4th Example which applied the optimization problem solution system. It is a figure which shows the 5th Example which applied the optimization problem solution system. It is a hardware block diagram which shows an example of the computer which realizes the function of the optimization problem solving apparatus.

Hereinafter, embodiments of the present disclosure will be described in detail with reference to the drawings. It should be noted that this embodiment does not limit the optimization problem solving method and the optimization problem solving device according to the present application. Further, in each of the following embodiments, duplicate description will be omitted by assigning the same reference numerals to the same parts.

The present disclosure will be described according to the order of items shown below.
1. 1. Technical overview 1-1. Background (applicable fields, etc.)
1-2. Sparse modeling 2. First Example 2-1. Device configuration of optimization problem solving system 2-2. Outline of processing of optimization problem solving system 2-2-1. Optimal selection of observation dimensions 2-2-2. L0 sparse modeling 2-2-3. User interface 2-2-4. Ising Machine 2-2-5. Sparse modeling 2-2-5-1. L0 sparse modeling, L1 sparse modeling 2-2-5-2. L0 sparse modeling as a combinatorial optimization problem 2-2-5-3. L0 sparse modeling using binary variables 2-2-5-4. L0 sparse modeling in QUAB format 2-2-5-5. K-bit quantized L0 sparse modeling 2-2-5-6. L0 sparse modeling by Ising model 2-2-6. Example of flow of L0 sparse modeling 2-2-7. L0 sparse modeling program example 2-3. Configuration of optimization problem solving device 2-4. Configuration of Ising machine 2-5. Compressed sensing 2-5-1. Combinatorial optimization problem 2-5-2. QUAB format 2-5-3. Optimization of compressed sensing by Ising model 2-5-4. Flow example of compressed sensing optimization 2-5-5. Program example of compressed sensing optimization 3. Other Examples 3-1. Second Example (RFID)
3-2. Third embodiment (extension of the second embodiment)
3-3. Fourth Example (Target Detection)
3-4. Fifth Example (Image Diagnosis)
3-5. Sixth Example (Other)
4. Other configuration examples, etc. 4-1. Other configuration examples 4-2. How to generate information used for processing programs and parameters 4-3. Others 5. Effect of this disclosure 6. Hardware configuration

[1. Technical overview]
Before explaining the present disclosure, an outline of the technical background relating to the present disclosure will be described. In the following technical description, the description of the prior art will be omitted as appropriate.

[1-1. Background (applicable fields, etc.)]
First, the fields of application of sparse modeling and compressed sensing related to the processing according to the present disclosure will be described. Sparse modeling and compression sensing are used in various fields such as MRI (Magnetic Resonance Imaging), radio telescopes, radar (Radar), and lidar (LIDAR: Light Detection and Ranging, Laser Imaging Detection and Ranging). ing.

For example, MRI is known as an early practical application of compressed sensing. Early MRI had the disadvantage that it took a long time to shoot, but by reducing the number of measurements by compressed sensing, the shooting time was shortened, which led to a significant reduction in the burden on patients and medical staff.

In addition, for example, in astronomical imaging using astronomical radio wave information measured by a radio observatory, there are few radio observatories (radio waves) based on the prior knowledge that there are few non-zero components (sometimes referred to as "non-zero components") in images. It is known that the detailed image of a black hole can be restored by super-resolution from the information obtained by the observatory array). Further, in the imaging of the surrounding environment using a radar, a rider, or the like, acquisition of an image having a higher resolution than the conventional one with super-resolution has been studied by the same principle.

[1-2. Sparse modeling]
By the way, the current sparse modeling is an approximate solution method (for example, L1 sparse modeling), and in order to obtain the same solution as the ideal sparse modeling (for example, L0 sparse modeling) where the highest accuracy is expected, the number of dimensions of observation is required. Certain conditions that are difficult to calculate must be met. Specifically, the RIP (Restricted Isometry Property) coefficient of the observation matrix must satisfy a certain condition, but it is difficult to calculate the RIP coefficient.

Therefore, in the current L1 sparse modeling, it is often the case that the evaluation on the order guarantees the equivalence with the true sparse modeling. However, in the evaluation by order, it was unclear how much the number of measurements could actually be reduced. In other words, it was difficult to guarantee whether the original image was really restored by technologies such as super-resolution and data restoration. Therefore, there has been a demand for sparse modeling that guarantees complete restoration of data easily.

As such sparse modeling, for example, using L0 sparse modeling itself or optimizing the observation dimension selection can be considered.

However, L0 sparse modeling is known as one of the combinatorial optimization problems that are difficult to optimize. In fact, no solution in polynomial time has been found at this time, and it has been difficult to optimize in practical calculation time as the scale increases.

In addition, the evaluation of the quality of the dimension selection of the observed variables includes a combinatorial explosion in which the method of evaluating whether the conditions for reaching a solution that can be completely reproduced from the observation data obtained by the dimension selection are satisfied. , It was difficult to evaluate with a practical calculation time.

Therefore, in the present disclosure, we provide a method to realize a combinatorial optimization problem that is difficult to optimize, such as L0 sparse modeling, by quantum calculation at a relatively high speed compared to performing a combinatorial search in a so-called general-purpose computer or the like. do.

Details will be described later, but to explain the outline, L0 sparse modeling is regarded as a combinatorial optimization problem of binary variables representing zero / non-zero components of sparse variables and bit representations obtained by quantizing sparse variables, and this combinatorial optimization is performed. The problem is converted into an Ising model, which is optimized at high speed by a sparse machine such as a quantum annealing machine (hereinafter also referred to as "quantum computer") or a quantum-conceived computer, and the solution of the optimum combination obtained (hereinafter referred to as "optimal solution"). ”) To calculate the sparse solution.

Therefore, this disclosure also provides a method for realizing the optimization of the dimension selection of observed variables in compressed sensing by quantum calculation at a relatively high speed.

Specifically, the dimension selection of observed variables is regarded as a combinatorial optimization problem, optimized by the Ising machine, and the matrix of dimension selection for sparsification and its synthesis is calculated from the solution of the obtained optimum combination. is doing.

By adopting such a configuration, for example, L0 sparse modeling is an ideal sparse modeling that is expected to have the highest accuracy. Therefore, unlike the conventional L1 sparse modeling, is it within the range to which the approximation of sparse modeling can be applied? Alternatively, the effort to verify that the solution is valid can be reduced to ensure the correct solution.

In addition, it is now possible to optimally select the dimensions of the observed variables instead of randomly selecting them, reducing the time and effort required to verify that the obtained solution is valid, and finding the correct solution. I can guarantee it.

First, the overall system configuration, etc. will be explained below, and then each process and application example will be explained.

[2. First Example]
[2-1. Device configuration of optimization problem solving system]
First, the configuration of the optimization problem solving system 1 shown in FIG. 1 will be described. FIG. 1 is a diagram showing a configuration example of the optimization problem solving system of the present disclosure. As shown in FIG. 1, the optimization problem solving system 1 includes an optimization problem solving device 100, a sensor 50, and a plurality of rising machines 10. The optimization problem solving device 100, the sensor 50, and the plurality of rising machines 10 are connected to each other via a predetermined communication network (network NT) so as to be communicable by wire or wirelessly. The optimization problem solving system 1 may include a plurality of optimization problem solving devices 100 and a plurality of sensors 50. Further, the optimization problem solving system 1 is not limited to the above, and may include various devices necessary for processing, for example, a device such as the measurement controller shown in block B6 of FIG.

The optimization problem solving device 100 uses the data sensed by the sensor 50 (first data) to generate an objective function for sparse modeling of the first data, and the coefficient related to the variable to be optimized in the objective function. It is an information processing device (computer) that transmits a matrix to the rising machine 10. For example, the variable to be optimized is a binary variable that distinguishes between a non-zero component and a zero component for each component of the sparse variable that models the first data. Further, for example, the variable to be optimized is each bit variable obtained by quantizing each component of the sparse variable that models the first data. Further, the optimization problem solving device 100 receives the calculation result (second data) of the Ising machine 10 and generates an optimum solution based on the second data.

The sensor 50 is a device that converts the observed phenomenon into an electric signal or data and outputs it. The sensor 50 has a function of transmitting the sensed data to the optimization problem solving device 100. The sensor 50 may be any sensor such as an image sensor, a sound sensor, an acceleration sensor, a position sensor, a temperature sensor, a humidity sensor, an illuminance sensor, and a pressure sensor. The optimization problem solving system 1 may include a device having a function as a sensor, such as an MRI device 500 (see FIG. 23), and details of this point will be described later.

The plurality of Ising machines 10 include Ising machines 10a, Ising machines 10b, and the like. When the Ising machine 10a, the Ising machine 10b, and the like are described without particular distinction, they are described as "Ising machine 10". The optimization problem solving system 1 may include a plurality of Ising machines 10a and a plurality of Ising machines 10b. The Ising machine 10 is a computer (computer) that uses the Ising model. The Ising machine 10 is a computer (hereinafter, also referred to as “combinatorial optimization machine”) that solves a combinatorial optimization problem using an Ising model.

The Ising machine 10a is a quantum computer (quantum computer) that solves a problem using quantum annealing. The Ising machine 10a can be realized by a quantum annealing method or a quantum annealing method. The Ising machine 10b is a quantum-conceived computer using CMOS (Complementary metal-oxide-semiconductor) or the like. For example, the rising machine 10b may be a quantum-conceived computer using a processor such as a GPU (Graphics Processing Unit) or an integrated circuit such as an FPGA (Field Programmable Gate Array). The Ising machine 10 is not limited to the Ising machine 10a and the Ising machine 10b described above, and may have any hardware configuration as long as it can solve the combinatorial optimization problem.

[2-2. Overview of optimization problem solving system processing]
Here, an outline of the processing of the optimization problem solving system 1 will be described with reference to FIG. FIG. 2 is a block diagram of the first embodiment.

FIG. 2 shows the overall configuration including L0 sparse modeling and observation dimension optimum selection. In the overall configuration diagram of FIG. 2, the upper block group (blocks B1 to B5 in FIG. 2) is a block that realizes the optimum selection of the observation dimension. On the other hand, the lower block group (blocks B6 to B10 in FIG. 2) is a block that realizes L0 sparse modeling. Each of the blocks B1 to B10 shown in FIG. 2 indicates any of the devices, data, and functions, and corresponds to any of the devices of the optimization problem solving system 1.

For example, the function of L0 sparse modeling shown in block B8 and the function of data generation shown in block B10 are executed by the optimization problem solving device 100. For example, the data of the sparse observation model shown in the block B5 may be possessed by the optimization problem solving device 100. For example, the data of the sparse observation model includes information on the selected dimension, information on the matrix of dimension compression, and the like. Further, for example, the combination optimization machine shown in block B3 and the combination optimization machine shown in block B9 correspond to the Ising machine 10. The combinatorial optimization machine shown in block B3 and the combinatorial optimization machine shown in block B9 may be the same Ising machine 10 or different Ising machines 10. For example, the combinatorial optimization machine shown in block B3 may be the Ising machine 10b as the first Ising machine, and the combinatorial optimization machine shown in block B9 may be the Ising machine 10a as the second Ising machine. ..

For example, the sensor indicating the block B7 corresponds to the sensor 50. Further, the measurement controller shown in the block B6 may be integrated with the sensor 50, or the sensor 50 may be included in the optimization problem solving system 1 as a separate device. The above is an example, and the correspondence between the blocks B1 to B10 shown in FIG. 2 and each device of the optimization problem solving system 1 may be any correspondence as long as the processing is feasible. ..

[2-2-1. Optimal selection of observation dimensions]
The blocks B1 to B5 in FIG. 2 corresponding to the observation dimension optimum selection are usually performed by offline processing, and are realized in advance regardless of the sensor input.

In the process of optimal selection of observation dimensions, as shown in block B1, which is a complete observation model, an observation matrix (also called "complete observation matrix") representing complete observations that is not sparse is input, and this observation dimension selection information is used. , An observation matrix representing sparse observations is output. The matrix representing the complete observation is, for example, a matrix representing the DFT (discrete Fourier transform), a matrix representing the wavelet transform, or a matrix acquired by dictionary learning or the like.

First, when a complete observation matrix that is not a sparse is input to the observation dimension selection modeling shown in block B2, combinatorial optimization of observation dimension selection is modeled based on the values of this complete observation matrix, and further, isging. The conversion to the coefficient matrix of the model is done. For example, the optimization problem solving device 100 models the combinatorial optimization of the observation dimension selection by the observation dimension selection modeling using the complete observation matrix, and generates the coefficient matrix of the Ising model.

Then, this coefficient matrix is input to the Ising machine 10 corresponding to the combinatorial optimization machine shown in the block B3, and the Ising machine 10 corresponding to the combinatorial optimization machine of the block B3 outputs a solution corresponding to the optimum combination. .. For example, the Ising machine 10 of the block B3 transmits a solution corresponding to the optimum combination to the optimization problem solving device 100.

In the optimum selection observation model shown in block B4, the selection information of the observation dimension and the sparse observation matrix (for example, the sparse corresponding to the selected observation dimension) are based on the information of the optimum combination such as the solution corresponding to the optimum combination. An observation matrix that represents various observations) is output. For example, the optimization problem solving device 100 selects an observation dimension based on the information of the optimum combination, and generates selection information of the observation dimension and a sparse observation matrix.

In the sparse observation model shown in block B5, the selection information of the observation dimension and the sparse observation matrix are saved in a file or the like so that they can be used in the measurement controller shown in block B6 and the L0 sparse modeling shown in block B8.

[2-2-2. L0 sparse modeling]
The blocks B6 to B10 in FIG. 2 corresponding to L0 sparse modeling are performed by online processing and are performed for the input of the sparse-observed sensor. It should be noted that the blocks B6 to B10 may be slightly modified to accumulate the input from the sensor, and then the subsequent processing may be realized by offline processing. In this case, the block B7 may be replaced with a data storage unit or the like, and the data storage unit of the block B7 may be included in the optimization problem solving device 100.

First, the measurement controller shown in block B6 controls the sensor shown in block B7 according to the measurement method read by the CPU (Central Processing Unit) or the like. For example, the measurement controller shown in the block B6 controls the sensor shown in the block B7 by using the information of the selection dimension included in the sparse observation model. It should be noted that the communication for transferring the sensor data may be controlled instead of the sensor shown in the block B7.

Subsequently, the data observed in the sparse by the sensor shown in the block B7 is input to the L0 sparse modeling shown in the block B8. It is assumed that the sparse observation matrix is input in advance to the L0 sparse modelin shown in the block B8. Based on the sparse observation data and the sparse observation matrix, modeling is performed as a combinatorial optimization of L0 sparse modeling, and further, the singing model is converted into a coefficient matrix. For example, the optimization problem solving device 100 performs modeling as a combinatorial optimization of L0 sparse modeling by L0 sparse modeling using the sparse observation data and the sparse observation matrix, and generates a coefficient matrix of the singing model.

Then, this coefficient matrix is input to the combinatorial optimization machine shown in block B9, and the Ising machine 10 corresponding to the combinatorial optimization machine of block B9 outputs a solution corresponding to the optimum combination. For example, the Ising machine 10 of the block B9 transmits a solution corresponding to the optimum combination to the optimization problem solving device 100.

In the data generation (restoration / synthesis) shown in block B10, the sparse solution is calculated based on the information of the optimum combination such as the solution corresponding to the optimum combination. For example, the optimization problem solving device 100 calculates a sparse solution based on the information of the optimum combination. Then, depending on the application to which it is applied, data is restored or generated based on the observation matrix of complete observation that has been input in advance.

[2-2-3. User interface]
Here, a user interface (hereinafter, may be referred to as “UI”) for a user who uses the optimization problem solving system 1 will be described.

(UI flow)
First, the processing flow related to the UI will be described with reference to FIG. FIG. 3 is a flowchart showing a processing procedure related to the user interface. In FIG. 3, the optimization problem solving device 100 provides a UI by means of an input unit 140, a display unit 150 (see FIG. 13), and the like, which will be described later. The case where the optimization problem solving device 100 provides the UI will be described as an example, but the present invention is not limited to the optimization problem solving device 100, and other devices such as the user's smartphone may provide the UI to the user. ..

As shown in FIG. 3, the optimization problem solving device 100 presents a list of combinatorial optimization tasks (step S1). For example, the optimization problem solving device 100 displays a list of a plurality of combinatorial optimization tasks such as measurement sparsification and sparsity modeling, as shown in the content CT1 in FIG.

Then, the optimization problem solving device 100 presents an optimization method that can be used (step S2). For example, as shown in the content CT1 in FIG. 4, the optimization problem solving device 100 displays a list of a plurality of optimization methods such as optimization by a quantum computer and optimization by a quantum-conceived computer. As shown in FIG. 1, a user presented with an Ising machine 10 for executing each optimization method selects an Ising machine 10 to be used from the presented Ising machines 10.

Then, the optimization problem solving device 100 presents the characteristics of the selected optimization method (step S3). For example, the optimization problem solving device 100 displays the characteristics of the selected optimization method as shown in the content CT2 in FIG.

Then, the optimization problem solving device 100 confirms whether or not the selected optimization method is acceptable (step S4). For example, the optimization problem solving device 100 displays a screen for confirming whether or not the selected optimization method is acceptable, as shown in the content CT3 in FIG.

Then, the optimization problem solving device 100 executes the optimization by the selected optimization method (step S5). When the user selects "Yes" of the content CT3 in FIG. 4, the optimization problem solving device 100 executes the process according to the task selected by the user and the optimization method.

As mentioned above, the optimization problem solving system 1 has two tasks to be optimized in combination, so the flow is configured so that the optimization method can be selected according to each. For example, the observation dimension optimization process, which may be offline process, requires a little communication time, but the optimization by the quantum computer on the cloud service can be selected. For example, the optimization problem solving system 1 may present to the user a method of optimization by a quantum computer (for example, a rising machine 10a) on a cloud service when the user selects measurement sparse as a task. .. When the user selects measurement sparsening as a task, the optimization problem solving system 1 may automatically select an optimization method by a quantum computer.

Also, for sparse modeling where online processing is desirable, optimization by a quantum-conceived computer on a local machine can be selected. For example, when the optimization problem solving system 1 selects sparse modeling performed by the user as a task, the optimization problem solving system 1 may present the user to be able to select the optimization method by the quantum-conceived computer on the local machine. When the user selects sparse modeling as a task, the optimization problem solving system 1 may automatically select an optimization method by a quantum-conceived computer (for example, a rising machine 10b) on a local machine. In this case, the optimization problem solving device 100 may be integrated with the Ising machine 10b.

Further, the optimization problem solving system 1 may automate the selection of tasks and optimization methods. For example, the optimization problem solving system 1 may automatically select the optimum combination of tasks and devices for the problem that the user wants to solve and present it to the user.

(UI image diagram)
Next, the image of the UI will be described with reference to FIG. FIG. 4 is a diagram showing an example of a user interface.

As shown in the content CT1 of FIG. 4, a UI (screen) prompting the user to select a task is first presented. In the example of FIG. 4, it is required to select either the observation dimension optimum selection (measurement sparsification) or sparse modeling, and the case where the “measurement sparsification” selected by the user is active is shown. In the case of a sparse modeling task, L1 sparse modeling may be selectable.

Further, the content CT1 in FIG. 4 shows an optimization method. Optimization methods include optimization processing by quantum annealing using a quantum computer, optimization processing using a quantum-conceived computer, simulated annealing, which is a general-purpose algorithm for combinatorial optimization, and integer programming solver (integer programming). It is possible to select from multiple candidates such as. For example, the optimization problem solving system 1 may include six Ising machines 10 that execute each of the six combinatorial optimization methods shown in the content CT1.

The content CT2 of FIG. 4, which is presented after the optimization method is selected, shows the characteristics of the optimization method. The content CT2 presents whether the selected optimization method is paid or free, the amount of money, and other specifications. Specifically, in the content CT2 in FIG. 4, the amount of money, the delay time, and the calculation time per Ising machine 10 that performs quantum annealing of the selected company D (may be a number of seconds different from the delay time). Etc. are included. Further, the content CT2 in FIG. 4 includes that the maximum number of variables is 1000 variables, the recommended number of joins is 6 or less, and the like. Further, the content CT2 in FIG. 4 includes various remarks such as recommended information indicating that the Ising machine 10 for performing quantum annealing of the selected company D is recommended for high-precision precalculation. The user selects an optimization method suitable for the purpose while observing the specifications and the like presented by the content CT2. The above UI presentation method is only an example, and the UI may be in any form as long as the user can select a desired task or optimization method.

[2-2-4. Ising machine]
Next, the Ising machine 10 will be described. An Ising machine is a machine that simulates the Ising model in hardware. The Ising model is a physical model originally proposed for modeling the phase transition phenomenon of a ferromagnet (magnet). For example, when the Ising model is represented by an illustration, the figure is as shown in FIG. FIG. 5 is a diagram schematically showing the Ising model.

The Ising model is a physical model proposed as a simple model for explaining the ferromagnetism of magnets. In FIG. 5, each grid point represents an electron spin in which only an upward or downward spin moment is observed.

The following equation (1) is a Hamiltonian (energy function) of the Ising model, which is a more generalized model of a simple magnet, and is also called a spin glass model.

_{Σ i in} equation (1) is a variable that corresponds to spin and takes either -1 or 1. Also, J _ij corresponds to the binding energy, and in the case of magnets, it occurs only between adjacent spins. Further, h _k corresponds to a local magnetic field and becomes a constant value in the case of a magnet.

When comparing the model shown in equation (1) with the objective function of the combinatorial optimization problem, which is often difficult to solve, they have an equivalent form. The following equation (2) is the objective function of the QUADO (Quadratic Unconstrained Binary Optimization) problem.

Here, the objective function shown in equation (2) is a Hamiltonian of the Ising machine, except for _{σ i} (same for σ _j and σ _{k), which is a binary variable that takes either 0 or 1.} It has the same format as 1). Therefore, instead of considering a difficult optimization algorithm, quantum annealing was devised with the idea of leaving the optimization to the realization of the physical ground state of the Ising machine.

In addition, J _ij or h _k may be called an Ising parameter or a coefficient of the Ising model, but it may be simply described as a parameter. _{For example, σ i} , σ _j , and σ _k in Eq. (2) correspond to the variables for which the optimum solution is to be found. Also, J _ij and h _k are parameters and are given from the outside. For example, the Ising machine 10 _{acquires the parameters J ij} and h _k from other devices such as the optimization problem solving device 100.

Quantum annealing deals with a model as shown in the following equation (3), which is the Ising model plus the term of quantum fluctuation.

At first, in the model shown in equation (3), the term of quantum fluctuation is dominated, the superposition state of all combinations is realized, and the Hamiltonian of the Ising model is gradually dominated. For example, the Ising machine 10 makes the second term on the right side dominant in the equation (3), and gradually makes the first term on the right side dominant.

The combination of spins eventually reaches the ground state, that is, the state of minimizing energy, provided that this process is adiabatic. Therefore, if the coefficient matrices J _ij and h _k of the QUABO problem are set in the Hamiltonian of the Ising model in advance, the final spin combination of the Ising model corresponds to the binary variable. In this way, by embedding the problem of combinatorial optimization in the Ising model and adiabatically transitioning from the state of quantum fluctuation to the ground state of the Ising model, it is possible to realize the solution of combinatorial optimization in the Ising spin with quantum annealing. Call.

For example, in equation (3), the superposition term is dominant at t = 0, and all combinations (solution candidates) are superposed. Then, in the equation (3), the superposition term is loosened step by step (for example, t is increased) to converge, and finally the ground state is realized. After that, the procedure is to measure the spin in the ground state and use this as the optimum solution. Since these procedures are realized by using the conventional technique of quantum annealing, detailed description thereof will be omitted.

In quantum annealing, it is said that relatively high-speed optimization is realized by smoothly passing through the energy barrier due to the quantum tunneling effect associated with quantum fluctuations as shown in Fig. 6. FIG. 6 is a diagram schematically showing the tunnel effect.

However, if quantum annealing is realized on an actual machine, it is difficult to maintain the quantum state, and this effect is said to be limited.

Therefore, while diverting the idea of embedding combinatorial optimization in the Ising model, dedicated hardware (for example, quantum-conceived type) that accelerates simulated annealing that minimizes energy by thermal fluctuation using a non-quantum device such as a transistor circuit. Computer 10b, etc.) has also been proposed.

As mentioned above, in this disclosure, hardware that simulates the Ising model, such as quantum computers and quantum-conceived computers, is collectively referred to as the Ising machine. That is, the Ising machine 10 is not limited to a quantum computer as long as it is hardware that simulates the Ising model, but is a device (quantum-inspired computer) or the like configured by using a processor such as a GPU or an integrated circuit such as an FPGA. You may.

What is common in realizing combinatorial optimization using these Ising machines, including quantum computers such as quantum annealing machines, is a quadratic form that can make the original combinatorial optimization problem equivalent to the Ising model. Write down on the binary optimization problem. This point will be described below.

[2-2-5. Sparse modeling]
[2-2-5-1. L0 sparse modeling, L1 sparse modeling]
From here, sparse modeling will be described. First, an outline of L0 sparse modeling and L1 sparse modeling will be described.

Given an N-dimensional observation variable x and an N × M-dimensional observation matrix U, the observation variable x is described as an M-dimensional vector z (simply “vector z”” as shown in the following equation (4). There is) to model.

In the case of N = M, if the observation matrix U has an inverse matrix, the solution of the model can be obtained by the following equation (5).

On the other hand, when N <M, the vector z is generally indefinite, and usually the solution cannot be uniquely obtained. However, if the number of non-zero components of the vector z is N or less, the vector z may be determined. A solution with few non-zero components is called a sparse solution, and modeling by a sparse solution is called sparse modeling. FIG. 7 is a conceptual diagram showing sparse modeling. In FIG. 7, the matrix D11 corresponds to the observation variable x, the matrix D12 corresponds to the observation matrix U, and the matrix D13 corresponds to the vector z. As shown in the matrix D12, in the observation matrix U, N indicating the number of rows is smaller than M indicating the number of columns. The white rectangles in the matrix D13 indicate the zero component (sometimes referred to as the "zero component"), and the hatched rectangles indicate the non-zero component. As shown in the matrix D13, it is known that the vector z for which the solution is to be obtained has sparse non-zero components. Such a solution in which the non-zero components of the vector z are sparse is also called a sparse solution z.

To actually find a sparse solution, solve the L0 norm minimization problem as shown in equation (6) below.

_{|| z || 0} in the equation (6) is called the L0 norm of the vector z, and represents the number of non-zero components of the vector z. In the present disclosure, this minimization problem will be referred to as L0 sparse modeling.

L0 sparse modeling is a combinatorial optimization problem, and if you try to solve it directly, it is necessary to thoroughly investigate all combinations. Therefore, L0 sparse modeling is not practical in the method of examining all combinations in a blunt manner as described above, and instead, the problem of minimizing the L1 norm as in the following equation (7) is often solved.

_{|| z || 1} in the equation (7) is called the L1 norm of the vector z, and is the sum of the absolute values of the components of the vector z. In the present disclosure, this minimization problem will be referred to as L1 sparse modeling.

Unlike L0 sparse modeling, L1 sparse modeling is not a combinatorial optimization problem and can be solved efficiently within the range of linear programming. Therefore, in sparse modeling, it is the de facto standard to obtain a sparse solution by L1 sparse modeling from the viewpoint of practicality.

By the way, the problem of sparse modeling was originally a problem for which a solution could not be determined. Then, the narrowing down of the solution is realized by assuming the sparsity of the solution. However, even if the sparsity of the solution can be assumed, it is not always the true solution. Therefore, even if the data is restored from the sparse solution or the super-resolution data is generated, there is no guarantee that the generated data is close to the true data.

For such a problem, in L0 sparse modeling, if any column vector that constitutes the observation matrix U by the number of non-zero components of the vector z is selected and these are linearly independent, it is the only true. Has been proven to reach the solution of. And it turns out that this condition is relatively easy to achieve.

On the other hand, in L1 sparse modeling, evaluation of the condition to reach the true solution, that is, the condition to reach the same solution as L0 sparse modeling requires evaluation of the RIP constant (RIP coefficient). The condition using the RIP constant is also called the RIP condition.

The RIP condition is a condition that, when expressed qualitatively, when any column vector constituting the observation matrix U is selected by the number of non-zero components of the vector z, these are in a state close to orthogonality. This condition is not always an easily achievable condition. Further, in the RIP condition, it is difficult to evaluate the RIP constant itself, and it is usually evaluated on the order of (complexity), and it is not possible to strictly evaluate whether the true solution is reached very accurately.

That is, the condition for reaching the true solution of L0 sparse modeling is "first-order independence", while the condition of L1 sparse modeling is "close to orthogonal", which is a little strict condition. Not only are L0 sparse modeling less constrained to the observation matrix than L1 sparse modeling, but they are also expected to be noise-robust and have a high compression ratio in terms of performance.

Therefore, there is a request to find a sparse solution using L0 sparse modeling. As described above, if it is a binary combinatorial optimization problem in a quadratic form, the Ising machine 10 can find a solution for combinatorial optimization at a relatively high speed.

Therefore, the optimization problem solving device 100 of the optimization problem solving system 1 re-formulates the L0 sparse modeling as a combinatorial optimization problem, and further converts the combinatorial optimization problem into a binary combinatorial optimization problem in a secondary format. ..

In the following, the process of formulating the optimization problem solving device 100 as a combinatorial optimization problem of L0 sparse modeling, converting the combinatorial optimization problem into a singing model, and calculating the sparse solution from the combinatorial optimization solution will be described.

[2-2-5-2. L0 sparse modeling as a combinatorial optimization problem]
First, the formulation as a combinatorial optimization problem of L0 sparse modeling will be described.

If the set of subscripts of the non-zero components of the sparse solution z is S (hereinafter also referred to as "set S"), it can be expressed as the following equation (8).

_{However, z S} in Eq. (8) is a vector composed only of S rows in the sparse solution z, and F _S is a matrix composed only in column S (S is a set) in the observation matrix U. _{For example, z S} in Eq. (8) is a vector composed of lines corresponding to each subscript included in the set S of the sparse solution z arranged in the order of the subscripts. In addition, F _S is a matrix composed of columns corresponding to each subscript included in the set S of the observation matrix U arranged in the order of the subscripts.

At this time, the sparse solution can be obtained as shown in the following equation (9).

_{However, W S} of the formula (9) can be obtained by the following formula (11) when the following formula (10) is satisfied.

Using the above, consider the penalty function shown in the following equation (12).

The penalty function in equation (12) is _{0 when z S} is a sparse solution, and is greater than 0 otherwise. Therefore, it is necessary for the set S to satisfy P (S) = 0. When this set S is used, L0 sparse modeling can be expressed as the following equation (13).

Here, | S | represents the number of elements of the set S, and is as shown in the following equation (14).

Λ is a hyperparameter that represents the weight of the penalty term, and set a large value to ensure that the constraint is satisfied.

Here, if the following equation (15) is used, the following equation (16) holds for the right side of the equation (12).

In this case, L0 sparse modeling can be expressed as the following equation (17).

[2-2-5-3. L0 sparse modeling using binary variables]
In the formulation up to the above equation (19), the set S of subscripts of non-zero components is obtained as the solution of combinatorial optimization. However, this formulation cannot be converted into the Ising model of the final goal as it is. Therefore, the optimization problem solving device 100 formulates using binary variables so that it can be handled by the Ising model. Therefore, consider a diagonal matrix B in which the components of the column S are 1 and the others are 0. For example, a diagonal matrix B is used in which the component of the column corresponding to each subscript included in the set S is 1 and the other elements are 0. Specifically, the diagonal matrix B as shown in the following equation (18) is used. The optimization problem solving device 100 uses a diagonal matrix B as shown in the equation (18). Equation (18) satisfies the following equation (19).

The diagonal component of the diagonal matrix B shown in equation (18) corresponds to a binary variable that determines whether the component is zero or non-zero for each dimension of the sparse solution. Using the diagonal matrix B shown in equation (18), the following equation (20) can be derived.

Focusing on the adjugate matrix equation of the component of the subscript i, where b _{i = 0, the following equation (21) first holds for the determinant.}

The relationship of equation (21) is shown by FIG. FIG. 8 is a conceptual diagram showing a determinant related to binary conversion. The matrix D21 in FIG. 8 shows a case where the determinant is 0. Further, the determinant D22 in FIG. 8 shows a case where the determinant is the determinant of the hatched portion. Further, the matrix D23 in FIG. 8 shows a matrix for setting the position where the diagonal component of ^{BF T FB is 0 to 1.}

Next, consider the Cramer's rule of the inverse matrix as shown in the following equation (22).

Since each component is a determinant, by sequentially applying the above relations of the determinant, the above formula (for example, equation (20)) can be obtained as a result.

Therefore, it can be expressed as the following equation (23).

Here, the following equation (24) is used for deriving the equation (23). For example, the optimization problem solving device 100 uses the equation (24) to generate the equation (23).

Then, L0 sparse modeling can be formulated as a binary combinatorial optimization problem as shown in the following equation (25).

Here, the following equation (26) was used to derive the equation (25). For example, the optimization problem solving device 100 uses the equation (26) to generate the equation (25).

As described above, the optimization problem solving device 100 can formulate the problem of L0 sparse modeling as a combinatorial optimization problem in which each dimension of the sparse solution is a binary variable. From the above, it can be seen that the problem of L0 sparse modeling is a combinatorial optimization problem in which each dimension of the sparse solution is a binary variable.

[2-2-5-4. L0 sparse modeling in QUABO format]
In order to handle a combinatorial optimization problem in the Zing model, it is necessary that the objective function of the combinatorial optimization problem is not only represented by a binary variable but also represented by a quadratic form of the binary variable. Therefore, in the equation (25), it is necessary to a position which is represented by the inverse matrix, such as ^{(I-B + BU T UB} ) -1, gradually being replaced with quadratic form.

Assuming that the absolute values of the eigenvalues of C = IU ^T U are all less than 1, the following equation (27) holds.

If the absolute value of the eigenvalues of C = IU ^T U is sufficiently smaller than 1, that is, if the columns of the observation matrix U are close to being orthogonal, the right-hand side can be approximated only by the term K = 1. When approximating with K = 1, it can be expressed as the following equation (28).

Thereby, the L0 sparse modeling is approximated by the combinatorial optimization problem as in the following equation (29).

This combinatorial optimization problem is in the quadratic form of _{binary variables (b 1} , b ₂ … b _M). Therefore, it can be treated as a QUAO problem. When approximating with K> 1, it can be expressed as the following equation (30).

Thereby, the penalty term is approximated as in the following equation (31).

The penalty term in equation (31) is in 2K dimensional form with respect to the _{binary variables (b 1} , b ₂ ... b _M). In order to treat it as a QUAO problem, it is necessary to reduce the 2K dimension to 2 dimensions. For this, the penalty term will be developed using a program or the like. For example, the optimization problem solving device 100 reduces the dimension of the penalty term to two dimensions by using a program that expands the penalty term.

Then, every time a term of the square of the binary variable or more appears, a new auxiliary binary variable is introduced and the term is reduced to the square of the auxiliary binary variable or less. For example, when the Lth-order term of a binary variable appears, the dimension is reduced to the first-order term by adding _{L auxiliary variables (b p1} … b _{pL) as shown in the following equation (32).} can.

However, this auxiliary variable must satisfy the following equation (33). Therefore, if this constraint is satisfied, it becomes 0, and if not, a penalty term such as the following equation (34), which takes a value larger than 0, is applied to the objective function (for example, the equation (for example,) each time the L-th order term appears. 29)).

When actually adding, add by applying the _{appropriate positive hyperparameter λ pm.} As a result, the optimization problem solving device 100 can convert to a QUABO problem, which is a quadratic form of a binary variable, even when approximating with K> 1. In this case as well, it can be expected that the smaller the eigenvalue of ^{IU T U, the better the accuracy of approximation.} That is, ^{it can be said that the closer U T} U is to the identity matrix, the better.

In addition, it is desirable that the eigenvalue (spectral norm) of the maximum absolute value of ^{IU T} U is 1 or less for the approximation of the series expansion and censoring of the inverse matrix.

However, if this condition does not always apply, it is possible to realize series expansion by incorporating the effect of regularization.

Specifically, consider optimization by adding the L2 regularization term to the original L0 optimization formula as shown in the following formula (35).

In this case, the inverse matrix is replaced by the following equation (36), and the eigenvalues are multiplied by 1 / (1 + γ) as a whole.

As a result, by adjusting the regularization, ^{it is possible to reduce the absolute value of the eigenvalues of (IU T} U) / (1 + γ) and realize the conditions that enable series expansion and truncation approximation of the inverse matrix. Become.

In the above example, a method of finding the optimum solution by approximating the inverse matrix is shown, but the optimization problem solving device 100 is not limited to the above method, and can be converted into a QUAO problem by various methods. good. This point will be described below.

For example, it is possible to eliminate the inverse matrix from the optimization formula by using auxiliary variables. The optimization problem solving device 100 may use an auxiliary variable to remove the inverse matrix from the optimization equation. Let us reconsider the second term of the optimization equation (25) including the inverse matrix described above. The second term of the equation (25) can be modified as the following equation (37).

Here, the auxiliary variable is introduced as shown in the following equation (38).

Here, the second equal sign uses Woodbury's inverse matrix formula.

Z in equation (38) is a vector with the same number of dimensions as the solution. Then, instead of the second term of the equation (25), as shown in the following equation (39), an objective function including the Lagrange undetermined constant term can be considered.

Here, w in equation (39) is a vector with the same number of dimensions as z. L0 sparse modeling can be expressed as the following equation (40) in the form including the Lagrange undetermined constant term of the equation (39).

This optimization can be solved by alternating optimization of B, z, and w. Alternate optimization is realized by the following pseudo algorithm.

(1-1) Initialize z and w (random initialization, zero initialization, etc.)
(1-2) Fix z and w and find B using the Ising machine (1-3) Fix b and update z and w For example, using the steepest descent method, the following equation (41) ), (42).

(1-4) Repeat (1-2) and (1-3) until the value of the optimization objective function converges or is executed a predetermined number of times.

The optimization problem solving device 100 may use the solution z obtained as described above as it is as a sparse solution z ^* . However, for this algorithm, z ^* may contain a numerical error. Therefore, the optimization problem solving device 100 may ^{use only B *} (b ^* ) and obtain a ^{sparse solution z *} again in the subsequent processing.

For L0 sparse modeling, as another solution using the Lagrange undetermined constant, a simpler formulation such as the following equation (43) may be used.

In this case as well, it can be solved by alternating optimization of B, z, and w. Alternate optimization is realized by the following pseudo-algorithm, which is a modification of the update equation of (1-3) of the above-mentioned algorithm.
(2-1) Initialize z and w (random initialization, zero initialization, etc.)
(2-2) Fix z and w and find B using Ising machine (2-3) Fix b and update z and w For example, using the steepest descent method, the following equation (44) ), (45).

(2-4) Repeat (2-2) and (2-3) until the value of the optimization objective function converges or is executed a predetermined number of times.

[2-2-5-5. K-bit quantized L0 sparse modeling]
Subsequently, as another method, K-bit quantized L0 sparse modeling will be described here.

To briefly restate L0 sparse modeling, it is formulated as a minimization problem of the objective function (loss function) as in the following equation (46).

Here, w in equation (46) is a Lagrange undetermined constant. The objective function of the equation (46) is often used when x does not contain noise, but when noise is included, the objective function (loss function) as in the following equation (47) is often used. ..

Here, λ is a positive constant determined according to the magnitude of noise. Although K-bit quantized L0 sparse modeling can be realized by using either the objective function of the equations (46) and (47), the case where the objective function of the equation (47) is used will be described first.

K-bit quantization In L0 sparse modeling, consider quantizing each component of the sparse solution z with K bits as shown in the following equation (48).

Here, s _i of equation (48) represents the sign of _{_{z i, (b i1, b}} i2 ... b iK-1) is | a binary representation of | z _i. s _i can also be replaced _{with the binary variable b i} by using the following equation (49).

In this case, it can be transformed as shown in the following equation (50).

Here, a _ik is a binary variable that satisfies the constraint condition of the equation (51).

Using equation (50), the second term in the objective function of equation (47) can be expressed in the quadratic form of a binary variable as in equation (52) below.

On the other hand, the first term in the objective function of equation (47) can be expressed as in equation (53) below.

This is because the series is 1 only in the case of the following formula (54), that is, in the case of the following formula (55), and 0 in the case of the following formula (56).

_{Here, a new binary variable c ik} that satisfies the constraint conditions shown in the following equations (57) and (58) is prepared.

Then, since it can be replaced with the following equation (59), the following equation (60) is obtained.

From the above, the objective function can be written down in the following equation (61) and in a quadratic form.

However, since the _{constraints on the auxiliary variables a ik} and c _ik are not taken into consideration, the correct solution cannot be obtained as it is.

Therefore, first, consider realizing the following equation (62), which is a constraint on _{the auxiliary variable a ik, in a quadratic form.}

The following equation (63) is a quadratic form that is 0 when the constraint shown in equation (62) is satisfied, and is larger than 0 otherwise.

Next, consider the following equation (64), which is a constraint on the auxiliary variable c _ik.

The following equation (65) is a quadratic form that is 0 when the constraint shown in equation (64) is satisfied, and is larger than 0 otherwise.

The objective function with these constraints (Equations (63) and (65)) can be expressed as the following equation (66).

It should be noted that the coefficients μ _ik and ν _ik of the constraint term may all be made common, and the following equation (67) may be used.

The objective function of equation (67) is a quadratic form of a binary variable, but when converted to a spin variable as shown in equations (68), (69), (70), and (71) below, it is quadratic of the spin variable. Expressed in form.

Since H (σ _a , σ _b , σ _c ) in equation (71) is a quadratic form of the spin variable, it can be optimized by the rising machine 10 and the optimum spin σ _a ^* , which minimizes the objective function. σ _b ^* and σ _c ^* can be obtained.

Obtain the obtained optimum spins σ _a ^* , σ _b ^* , σ _c ^* optimized binary variables a ^* , b ^* , c ^* as shown in the following equations (72), (73), and (74). Can be done.

Using the results of a ^* and b ^* , the sparse solution quantized by K bits can be obtained as shown in the following equation (75).

If higher resolution is required, it is possible to obtain ^{z *} again from the information assuming that only the information on whether the component of the solution is zero or non-zero is output. The zero / non-zero information of the component z _i ^* of the sparse solution is given by ^{c *} _ik-1.

The above is sparse modeling when there is noise, but in the formulation when there is no noise, the objective function of the following equation (76) is used instead of _{L λ (a, b, c).}

In this case, the objective function is a quadratic form of a binary variable, but it is necessary to add a constraint term as in the following equation (77).

Since the constraint term is also in the quadratic form, it is in the quadratic form of the binary variable as a whole, so that it can be solved by the Ising machine 10 at a relatively high speed.

However, since w is an undetermined constant, it is also necessary to estimate w. To do this, the following quantum annealing and alternating optimization of the gradient method are repeated until certain conditions are met.
(3-1) Initialize w (random number or zero initialization may be used)
(3-2) Optimize the combination variables b and c by the following equation (78) by quantum annealing.

(3-3) Update w by the gradient method by the following equation (79).

Here, η in the equation (79) is a learning rate parameter of the gradient method, and is set to 0.01, for example.
(3-4) Repeat (3-2) and (3-3) until a predetermined condition is satisfied, and set the obtained b and c as optimized binary variables b ^* and c ^* .

The sparse solution z ^* may be output from the binary variables b ^* and c ^* obtained as described above, or the zero-non-zero information c ^* _ik-1 of the sparse solution may be output.

The two methods have been explained above, but there are various other possible variations of K-bit quantized L0 sparse modeling. For example, it is conceivable that the quantization intervals are not equalized, or that the quantization intervals are increased as the absolute value increases. As for the L0 regularization term, since the lower bits have an error, the L0 regularization term may be represented only by the upper bits.

Further, for example, when it is known that the component of the sparse solution is 0 or more, the binary variable a may be omitted from the formulation. Further, as a method similar to the group Lasso, a case where the sparse solution becomes zero or non-zero in group units can be considered. When each component of the N sparse solution is further composed of L elements and all L elements are 0, it is considered as a zero component, and if not, it is considered as a non-zero component. In this case, the L0 regularization term can be written down as in the following equation (80).

_{Here, a new binary variable di l} that satisfies the constraint conditions shown in the following equations (81) and (82) is prepared.

Then, since it can be replaced with the following equation (83), the following equation (84) is obtained.

In order for the binary variable d _il to satisfy the constraint, the constraint term in the following equation (85) is added to the objective function.

[2-2-5-6. L0 sparse modeling with Ising model]
As described above, the optimization problem solving device 100 formulates the L0 sparse modeling as a combination optimization problem of binary variables, and further converts it into a quadratic form QUAS problem. In this way, it is easy to embed the QUABO problem converted by the optimization problem solving device 100 in the Ising model.

First, the optimization problem solving device 100 converts a binary variable into a spin variable in order to convert it into an Ising model. For this, the following equation (86) is used.

The optimization problem solving device 100 transforms the QUABO problem into a quadratic form of spin variables, that is, an Ising model, by the equation (86). Then, for the Ising model formulated in this way, the optimization problem solving device 100 extracts a coefficient matrix by using a program or the like. For example, the coefficient matrix is an array composed of coefficients related to the first-order or higher terms of the variables to be optimized extracted from the objective function. As described above, the optimization problem solving device 100 extracts the coefficient matrix as the singing parameter from the objective function corresponding to the first data from the sensor. Then, the optimization problem solving device 100 uses the extracted coefficient matrix to set the spin-to-spin coupling constant of the riser machine 10 and the local magnetic field. The optimization problem solving device 100 transmits the extracted coefficient matrix (Ising parameter) to the Ising machine 10. Then, the Ising machine 10 performs an annealing process, calculates a combination of basal spins of the Ising model (σ ^* ₁ ... σ ^* _M ...), and transmits the combination to the optimization problem solving device 100. The optimization problem solving device 100 receives the combination of basal spins of the Ising model calculated by the Ising machine 10.

The optimization problem solving device 100 can obtain a binary variable optimized from this combination of basis spins by using the following equation (87).

Therefore, the optimization problem solving device 100 obtains ^{a set S *} showing the combination of the subscripts of the optimized sparse components by selecting i having the following equation (88).

By using this set S ^* , the following equation (89) can be obtained.

The final sparse solution z ^* can be obtained by setting the element of the dimension of the set S ^* to Eq. (89) and the other dimensions to 0. The optimization problem solving device 100 calculates the element of the dimension of the ^{set S *} by the equation (89), and sets the other dimensions to 0 to generate the ^{final sparse solution z *.}

[2-2-6. L0 sparse modeling flow example]
Next, the flow of L0 sparse modeling will be described with reference to FIG. FIG. 9 is a flowchart showing the procedure of the processing of L0 sparse modeling. FIG. 9 is an example of the flow of L0 sparse modeling by the optimization problem solving system 1 including the Ising machine 10. In the following, a case where the optimization problem solving device 100 is the main processing element is shown as an example, but the device is not limited to the optimization problem solving device 100, and may be any device included in the optimization problem solving system 1.

As shown in FIG. 9, the optimization problem solving device 100 inputs observation data and an observation matrix (step S101). For example, the optimization problem solving device 100 acquires the observation data x and the observation matrix U by the sensor 50. Step S101 is a process corresponding to the above equation (4) and the like.

The optimization problem solving device 100 acquires the coefficient matrix of the combination formulation of the non-zero components of the solution (step S102). For example, the optimization problem solving device 100 generates a coefficient matrix of the combination formulation of the non-zero components of the solution as an singing parameter. The optimization problem solving device 100 extracts a coefficient matrix in the formulation of combinatorial optimization. In step S102, processing from the above-mentioned L0 sparse modeling to the QUABO problem and the formulation into the Ising model is performed. Step S102 is a process corresponding to the above equation (25), equation (29), and the like.

The optimization problem solving device 100 combines the coefficient matrices and sends them to the optimization machine (step S103). For example, the optimization problem solving device 100 transmits a coefficient matrix of the combination formulation of the non-zero components of the solution as an Ising parameter to the Ising machine 10 selected by the user.

The optimization problem solving device 100 receives the optimization solution from the combinatorial optimization machine (step S104). For example, the optimization problem solving device 100 receives the solution calculated by the Ising machine 10 as an optimized solution from the Ising machine 10 that has transmitted the Ising parameter. In this way, the optimization problem solving system 1 obtains a combination optimized by the Ising machine 10, which is a combinatorial optimization machine.

The optimization problem solving device 100 calculates the value of the non-zero component of the solution (step S105). For example, the optimization problem solving device 100 calculates the value of the non-zero component of the solution using the optimized solution received by the Ising machine 10. The optimization problem solving device 100 calculates a sparse solution based on the combination information of the sparse components. Step S105 is a process corresponding to the above equations (9) to (11) and the like.

The optimization problem solving device 100 restores and synthesizes data (step S106). For example, the optimization problem solving device 100 restores or synthesizes data according to the application to which it is applied, using the value of the non-zero component of the calculated solution.

[2-2-7. L0 sparse modeling program example]
From here, an example of a program (function) of L0 sparse modeling is shown with reference to FIGS. 10 to 12. FIG. 10 is a diagram showing an example of an L0 sparse modeling program. Specifically, FIG. 10 shows program PG1 which is an example of the whole program up to the generation of the solution of L0 sparse modeling.

The program PG1 shown in FIG. 10 is mainly composed of the following blocks.
-Generate spins and describe the objective function of L0 sparse modeling-Extract the Ising parameter (first-order and quadratic coefficient matrix) from the objective function-Send the Isging parameter to the Issing machine and receive the optimal spin arrangement-Optimal spin Calculate the solution of L0 sparse modeling from the variables

For example, the function "load_observation_matrix (file)" on the first line of the program PG1 reads the file and sets the observation matrix (U in FIG. 11). For example, the function "Spin (N)" on the second line of the program PG1 sets the variables supported by the library. For example, the function "IsingMachine ()" on the third line of the program PG1 sets the machine supported by the library.

For example, the function "l0_sparse_modeling (U, x, s, lambda = xxx)" on the fourth line of the program PG1 calculates the objective function (loss function, also referred to as loss) of L0 sparse modeling as shown in FIG. ..

FIG. 11 is a diagram showing an example of a program that generates an objective function of the Ising model. Specifically, FIG. 11 shows a program PG2 which is an example of a function for calculating an objective function of a QUABO problem (Ising model). The function "l0_sparse_modeling (U, x, s, lambda)" shown in FIG. 11 is called by the entire program PG1 of FIG. 10 to generate an objective function (corresponding to loss in FIGS. 10 and 11).

The program PG2 shown in FIG. 11 is mainly composed of the following blocks.
・ Create a binary variable that represents a non-zero component using a spin variable ・ Create an objective function that increases as the number of non-zero components increases ・ Create a penalty that increases when the observed observation and the expected observation from the sparse solution deviate from each other Add the objective function and the penalty to get the final objective function

For example, the function "diag (b = (1-s) / 2)" on the first line of the program PG2 converts the spin into a binary variable. For example, the function "trace (B)" on the second line of the program PG2 calculates the number of non-zero components. For example, the function "matmul (B, matmul (U.t, x))" on the third line of the program PG2 extracts components from the data. For example, the function "UnitMatrix-matmul (U.t, U)" on the fourth line of the program PG2 finds the deviation of the observation matrix from the unit matrix. For example, the function "matmul (D, matmul (Fx, Fx.t))" on the fifth line of the program PG2 sets a penalty term. For example, the expression on line 6 of program PG2 adds a penalty to the objective function.

The parameter λ (corresponding to lambda in FIG. 11) may be set from 0.5 to 1, or the spin (corresponding to s in FIG. 11) and the parameter λ (corresponding to lambda in FIG. 11). Alternate optimization may be performed to obtain the Lagrange undetermined multiplier method.

For example, the function "get_ising_params (loss)" on the fifth line of the program PG1 is a function supported by the library, and calculates a coefficient matrix (Ising parameter) by inputting an objective function (corresponding to loss in FIG. 11). ..

For example, the function "ising_macine.solve_ising (h, J)" on the sixth line of the program PG1 is a function supported by the library, and the calculation of the Ising machine 10 is performed by inputting a coefficient matrix (h, J in FIG. 11). Get results.

For example, the function "l0_sparse_solution (U, x, s_optimum)" on the 7th line of the program PG1 calculates the solution of L0 sparse modeling as shown in FIG.

FIG. 12 is a diagram showing an example of a program that generates a solution for L0 sparse modeling. Specifically, FIG. 12 shows a program PG3 which is an example of a function for obtaining a sparse solution from a combination variable. The function "l0_sparse_solution (U, x, s_optimum)" shown in FIG. 12 is called by the whole program PG1 of FIG. 10 to generate a sparse solution z.

The program PG3 shown in FIG. 12 is mainly composed of the following blocks.
-Use the selected rows to generate a submatrix of the observation matrix-Get a sparse solution from the pseudoinverse of the submatrix

For example, the formula in the first row of the program PG3 generates a submatrix of the observation matrix. For example, the function "matmul (Fs.t (), Fs)" on the second row of the program PG3 generates a square matrix. For example, the function "matmul (FstFs.i (), Fs.t ())" on the third line of the program PG3 generates a pseudo-inverse matrix. For example, the function "matmul (Ws, x)" on the fourth line of the program PG3 calculates the non-zero component of the sparse solution. For example, the function "ZeroVector ()" on the fifth line of the program PG3 generates the zero component of the sparse solution. For example, the equation on line 6 of program PG3 combines the zero component of a sparse solution with the non-zero component of a sparse solution to generate a solution.

For example, the optimization problem solving device 100 stores a program (function) as shown in FIGS. 10 to 12 and a program (function) called by each program in the function information storage unit 122 (see FIG. 13), and each of them is stored. Execute the process using the program.

[2-3. Configuration of optimization problem solving device]
Next, the configuration of the optimization problem solving device 100 will be described. FIG. 13 is a diagram showing a configuration example of the optimization problem solving device of the present disclosure.

As shown in FIG. 13, the optimization problem solving device 100 includes a communication unit 110, a storage unit 120, a control unit 130, an input unit 140, and a display unit 150.

The communication unit 110 is realized by, for example, a NIC (Network Interface Card) or the like. Then, the communication unit 110 is connected to a predetermined network (not shown) by wire or wirelessly, and transmits / receives information to / from other information processing devices such as the rising machine 10 and the sensor 50. Further, the communication unit 110 may send and receive information to and from a user terminal (not shown) used by the user.

The storage unit 120 is realized by, for example, a semiconductor memory element such as a RAM (Random Access Memory) or a flash memory (Flash Memory), or a storage device such as a hard disk or an optical disk. As shown in FIG. 13, the storage unit 120 according to the first embodiment has a data storage unit 121 and a function information storage unit 122.

The data storage unit 121 stores various data such as data received from the sensor 50. The function information storage unit 122 stores information on functions such as various programs. For example, the function information storage unit 122 stores information on various functions (programs) used in the optimization problem solving process and the generation process. For example, the function information storage unit 122 stores function programs such as programs PG1 to PG3. Further, the function information storage unit 122 may store information on the function used for processing among the above-mentioned expressions. The storage unit 120 is not limited to the above, and may store various information depending on the purpose.

In the control unit 130, for example, a program (for example, an optimization problem solving processing program according to the present disclosure) stored in the optimization problem solving device 100 by a CPU, an MPU (Micro Processing Unit), or the like is a RAM (Random Access). It is realized by executing Memory) etc. as a work area. Further, the control unit 130 is realized by, for example, an integrated circuit such as an ASIC (Application Specific Integrated Circuit) or an FPGA (Field Programmable Gate Array).

As shown in FIG. 13, the control unit 130 includes a reception unit 131, a first generation unit 132, a second generation unit 133, a transmission unit 134, and a third generation unit 135, which will be described below. Realize or execute the functions and actions of information processing. The internal configuration of the control unit 130 is not limited to the configuration shown in FIG. 13, and may be any other configuration as long as it is configured to perform information processing described later.

The receiving unit 131 receives various information. The receiving unit 131 receives various information from an external information processing device. The receiving unit 131 receives data from the sensor 50. The receiving unit 131 functions as a sensor input unit for inputting the first data from the sensor 50.

The receiving unit 131 receives various information from the Ising machine 10. The receiving unit 131 receives the calculation result of the Ising machine 10 from the Ising machine 10. The receiving unit 131 receives the second data as a calculation result from the Ising machine 10 to which the transmitting unit 134 has transmitted the coefficient matrix.

The first generation unit 132 generates an objective function by performing sparse modeling. The first generation unit 132 acquires information from the data storage unit 121 and the function information storage unit 122, and generates an objective function by performing sparse modeling using the acquired information. For example, the first generation unit 132 generates an objective function by the program PG2.

The second generation unit 133 generates a coefficient matrix corresponding to the objective function. The second generation unit 133 acquires information from the function information storage unit 122, and uses the acquired information and the objective function generated by the first generation unit 132 to generate a coefficient matrix corresponding to the objective function. For example, the second generation unit 133 generates a coefficient matrix corresponding to the objective function by the function "get_ising_params (loss)" of the program PG1.

The transmission unit 134 transmits various information to other information processing devices such as the Ising machine 10 and the sensor 50.

The transmission unit 134 instructs the Ising machine 10 to execute the calculation. The transmission unit 134 transmits the parameters of the Ising model to the Ising machine 10. The transmission unit 134 instructs the Ising machine 10 to execute the calculation by transmitting the parameters of the Ising model to the Ising machine 10. The transmission unit 134 transmits the coefficient matrix generated by the second generation unit 133 to the Ising machine 10 that performs the combinatorial optimization calculation.

The third generation unit 135 generates the optimum solution. The third generation unit 135 generates a solution using the calculation result of the Ising machine 10 acquired from the Ising machine 10. The third generation unit 135 uses the second data received from the Ising machine 10 by the reception unit 131 to generate a solution corresponding to the first data. For example, the third generation unit 135 generates an objective function by the program PG3.

Various operations are input from the user to the input unit 140. The input unit 140 accepts input by the user. The input unit 140 may accept various operations from the user via the keyboard, mouse, or touch panel provided in the optimization problem solving device 100. The input unit 140 accepts the user's selection of the Ising machine 10. The input unit 140 accepts the user's operation on the contents CT1 to CT3, which are the presentation screens displayed by the display unit 150, as input.

The display unit 150 displays various information. The display unit 150 is a display device such as a liquid crystal display and displays various information. The display unit 150 displays the contents CT1 to CT3 and the like. The optimization problem solving device 100 may have a content generation unit that generates contents CT1 to CT3 and the like. The content generation unit generates information to be displayed on the display unit 150, such as the contents CT1 to CT3. The content generation unit generates contents CT1 to CT3 and the like by appropriately using various technologies such as Java (registered trademark). The content generation unit may generate content CT1 to CT3 or the like based on the format of CSS, Javascript (registered trademark), or HTML. Further, for example, the content generation unit may generate contents CT1 to CT3 and the like in various formats such as JPEG (Joint Photographic Experts Group), GIF (Graphics Interchange Format), and PNG (Portable Network Graphics).

[2-4. Ising machine configuration]
Next, the configuration of the Ising machine 10 that executes the calculation will be described. FIG. 14 is a diagram showing a configuration example of the Ising machine. In the example of FIG. 14, the configuration of the Ising machine 10a, which is a quantum computer, will be described as an example of the Ising machine 10.

As shown in FIG. 14, the Ising machine 10a has a communication unit 11, a storage unit 12, a quantum device unit 13, and a control unit 14. The Ising machine 10a has an input unit (for example, a keyboard, a mouse, etc.) that receives various operations from the administrator of the Ising machine 10a, and a display unit (for example, a liquid crystal display, etc.) for displaying various information. You may.

The communication unit 11 is realized by, for example, a NIC or a communication circuit. The communication unit 11 is connected to a predetermined network (Internet or the like) by wire or wirelessly, and transmits / receives information to / from other devices such as the optimization problem solving device 100 via the network.

The storage unit 12 is realized by, for example, a semiconductor memory element such as a RAM or a flash memory, or a storage device such as a hard disk or an optical disk. The storage unit 12 stores various information used for displaying the information.

The quantum device unit 13 executes various quantum calculations. For example, the quantum device unit 13 is realized by a quantum processing unit (QPU: Quantum Processing Unit). The quantum device unit 13 realizes the ground state of the Ising model based on the parameters of the Ising model received from other devices such as the optimization problem solving device 100, for example. In other words, the quantum device unit 13 realizes the optimum spin arrangement in which the Ising model is in the base energy state. That is, the quantum device unit 13 realizes a state in which the optimization problem is optimized.

The quantum device unit 13 is composed of, for example, a plurality of qubits. The quantum device unit 13 is cooled to near absolute zero in advance. After the parameters of the Ising model are input to the quantum device unit 13, the quantum device unit 13 internally develops the ratio of the Ising model and the transverse magnetic field model (quantum fluctuation model) over time. As a result, the optimum spin arrangement according to the parameters of the Ising model is realized on the quantum device unit 13. In this way, the optimum spin arrangement of the Ising model is physically realized on the quantum device unit 13. Then, by measuring the quantum device unit 13, the optimum spin arrangement of the Ising model can be obtained. As a result, the quantum device unit 13 can optimize the discrete optimization problem. For example, the quantum device unit 13 can optimize the optimization problem of the objective function in the quadratic form.

The control unit 14 is realized by, for example, a CPU, an MPU, or the like executing a program stored in the Ising machine 10a with a RAM or the like as a work area. Further, the control unit 14 is a controller, and may be realized by an integrated circuit such as an ASIC or FPGA.

As shown in FIG. 14, the control unit 14 has an acquisition unit 141, a calculation unit 142, and a transmission unit 143, and realizes or executes the functions and operations of information processing described below. The internal configuration of the control unit 14 is not limited to the configuration shown in FIG. 14, and may be any other configuration as long as it is configured to perform information processing described later.

The acquisition unit 141 receives various information. The acquisition unit 141 receives various information from an external information processing device. The acquisition unit 141 receives various information from other information processing devices such as the optimization problem solving device 100.

The acquisition unit 141 performs a calculation using, for example, the quantum device unit 13, and receives an instruction for measurement from another information processing device such as the optimization problem solving device 100. The acquisition unit 141 accepts the parameters of the Ising model as instructions for calculation (measurement) by the quantum device unit 13.

The acquisition unit 141 acquires various types of information. The acquisition unit 141 acquires information from the storage unit 12. The acquisition unit 141 acquires various information from an external information processing device such as the optimization problem solving device 100. The acquisition unit 141 acquires the input information received by the input unit. The acquisition unit 141 acquires information about the parameters of the Ising model from, for example, an external information processing device. The acquisition unit 141 acquires the measurement result (calculation result) of the quantum device unit 13 by the calculation unit 142.

For example, the acquisition unit 141 receives the coefficients h and J (Ising parameters) of the Ising model, which is the processing result of the function of get_ising_params (loss) of the program PG1, from the optimization problem solving device 100.

The calculation unit 142 executes various calculations. The calculation unit 142 executes a calculation using the quantum device unit 13. The calculation unit 142 measures the quantum device unit 13. The calculation unit 142 measures the quantum device unit 13 in which the optimum spin arrangement of the Ising model is realized.

For example, the calculation unit 142 executes the calculation using the Ising parameter received by the acquisition unit 141 from the optimization problem solving device 100.

The transmission unit 143 transmits various information to an external information processing device. For example, the transmission unit 143 transmits various information to another information processing device such as the optimization problem solving device 100. The transmission unit 143 transmits the information stored in the storage unit 12.

The transmission unit 143 transmits various information based on information from other information processing devices such as the optimization problem solving device 100. The transmission unit 143 transmits various information based on the information stored in the storage unit 12.

The transmission unit 143 transmits the measurement result of the quantum device unit 13 by the calculation unit 142 to the device that has given the calculation instruction. The transmission unit 143 transmits the measurement result of the quantum device unit 13 by the calculation unit 142 to the parameter transmission source. The transmission unit 143 transmits the measurement result of the quantum device unit 13 by the calculation unit 142 to the request source for the calculation. The transmission unit 143 transmits the measurement result of the quantum device unit 13 by the calculation unit 142 to another information processing device such as the optimization problem solving device 100.

For example, in the example of FIG. 1, the transmission unit 143 transmits the value of the rising spin calculated (measured) using the parameters received from the optimization problem solving device 100 to the optimization problem solving device 100.

[2-5. Compressed sensing]
The compressed sensing to which the optimization problem solving method by the optimization problem solving system 1 is applied will be described below.

In compressed sensing, it is assumed that the observation matrix U (hereinafter also referred to as sparse observation matrix) can be decomposed as shown in the following equation (90).

Here, A in equation (90) is an observation matrix (also referred to as non-sparse observation matrix A) that models all observations.

Using the non-sparse observation matrix A and the sparse representation z, the observation y without dimensional decimation can be expressed by the following equation (91).

For example, the non-sparse observation matrix A is a discrete Fourier transform matrix or the like. The non-sparse observation matrix A is determined by the sensing mechanism and generally cannot be controlled.

On the other hand, C (also called matrix C) in equation (90) is a controllable variable. The matrix C is introduced for the purpose of reducing the observation dimension of equation (91).

[2-5-1. Combinatorial optimization problem]
From here, I will explain the point that compression is regarded as a combinatorial optimization problem. The same points as described above will be omitted as appropriate.

In L0 sparse modeling, the correlation of each column of the observation matrix U is small, and the closer it is to a state orthogonal to each other, the better the approximation accuracy of L0 sparse modeling. Therefore, as the matrix C, it is desirable to use a matrix that is transformed so that the correlation of each column of the equation (90) becomes small. Further, it is desirable that the matrix C can reduce the number of observations by reducing the dimension of observation.

The subscripts of the rows selected from the non-sparse observation matrix A are collectively referred to as the set S, the matrix A _S is a matrix composed of only the S rows of the non-sparse observation matrix A, and the matrix C _S is only the S column of the matrix C. It is assumed that it is a matrix composed of. For example, the matrix A _S is a matrix composed of rows corresponding to each subscript included in the set S of the non-sparse observation matrix A arranged in the order of the subscripts. Further, the matrix C _S is a matrix composed of columns corresponding to each subscript included in the set S of the matrix C arranged in the order of the subscripts. In this case, the following equation (92) is obtained.

The ^{degree of deviation of U T} U from the orthogonal matrix can be defined by the following equation (93).

Here, || · || _F is the Frobenius norm, which is the squared norm of all components. It ^{is 0 when U T} U matches the orthogonal matrix, and this value increases qualitatively as the correlation of each column increases. Therefore, the index shown in Eq. (93) can be regarded as the degree of deviation from the orthogonal matrix.

_{Here, the matrix C S} for minimizing the following equation (94) satisfies the following equation (95).

Multiplying both sides of the equation (94) from the left and right by the pseudo-inverse matrix (assuming it exists) gives the following equations (96) and (97).

Since the matrix C _S is an invertible matrix, the following equation (98) may be used.

Here, since A _S A _S ^T is a symmetric matrix, it can be diagonalized as in the following equation (99).

It is assumed that it is diagonalized as in equation (99). In this case, the matrix C _S is Eq. (100).

_{Using the matrix C S} of equation (100), the minimum value of the deviation from the orthogonal matrix is represented by the following number (101).

Therefore, the optimization of compression can be considered as a combinatorial optimization problem as shown in the following equation (102). The optimization problem solving device 100 can formulate the optimization of compression as a combinatorial optimization problem as in the equation (102).

Note that this formulation is a formulation that fixes the observation dimension and selects the optimum dimension among them, but there may be other formulations such as when you want to reduce the observation dimension as much as possible. In this case, the second term may be | S |.

Here, the following equation (103) is used as in the case of L0 sparse modeling.

In the optimization of compressed sensing from the relational expression of the equation (103), the optimization problem solving device 100 can also be formulated by the binary combinatorial optimization problem as in the following equation (104).

[2-5-2. QUABO format]
Regarding B in equation (104), I would like to reduce it to a quadratic form. For example, the optimization problem solving device 100 converts the equation (104) into a quadratic form of B.

For example, when the non-sparse observation matrix A is a discrete Fourier transform, the condition shown in the following equation (105) is satisfied.

In the case of the formula (105), it can be simply formulated as the following formula (106).

Further, when the eigenvalue of D = I-AA ^T is smaller than 1, it can be expressed by the following equation (107).

Then, when approximating with K = 1, it can be converted as shown in the following equation (108).

Equation (108) is quaternary with respect to B. However, L0 sparse modeling, [2-2-5-4. By the same process as the dimension reduction described in [L0 sparse modeling in QUAB format] and the like, the dimension can be easily reduced and converted into a quadratic expression. As described above, since it is the same as the process in the above-mentioned L0 sparse modeling, a detailed description thereof will be omitted, but for example, the optimization problem solving device 100 reduces the dimension B of the equation (108) to two dimensions.

For approximation of series expansion and censoring of the inverse matrix, it is desirable that the eigenvalue (spectral norm) of the maximum absolute value of ^{D = I-AA T is 1 or less.}

As with the L0 sparse model, it is possible to realize series expansion by incorporating the effect of regularization. By adding a regularization term to the original optimization equation, the inverse matrix is replaced by the following equation (109), and the eigenvalues are multiplied by 1 / (1 + γ) as a whole.

As a result, by adjusting the regularization, ^{it is possible to reduce the absolute value of the eigenvalues of (I-AA T} ) / (1 + γ) and realize the conditions that enable series expansion and truncation approximation of the inverse matrix. It becomes.

For example, as in the example of L0 sparse modeling, the inverse matrix can be eliminated from the optimization formula by using auxiliary variables. Using the following equation (110), the second term of the optimization equation (104) including the inverse matrix described above will be considered again.

Here, the auxiliary variable is introduced as shown in the following equation (111).

The second equal sign uses Woodbury's inverse matrix formula.

Here, Z in Eq. (111) is a matrix of the same size as the non-sparse observation matrix A. Then, instead of the first term of the equation (104), as shown in the following equation (112), an objective function including the following Lagrange undetermined constant term can be considered.

Here, W in equation (112) is a matrix of the same size as Z. It can be expressed as the following equation (113) in a format including the Lagrange undetermined constant term of this equation (112).

This optimization can be solved by alternating optimization of B, Z, and W. Alternate optimization is realized by the following pseudo algorithm.
(4-1) Initialize Z and W (random initialization, zero initialization, etc.)
(4-2) Fix Z and W and find B using the Ising machine (4-3) Fix B and update Z and W For example, using the steepest descent method, the following equation (4-3) 114), (115) and so on.

(4-4) Repeat (4-2) and (4-3) until the value of the optimization objective function converges or the execution is performed a predetermined number of times.

[2-5-3. Optimization of compressed sensing by Ising model]
If the optimization of compressed sensing is formulated as a combination optimization problem of binary variables as described above and further converted into a quadratic form QUABO problem, it is easy to embed this QUABO problem in the Ising model.

First, the optimization problem solving device 100 converts a binary variable into a spin variable in order to convert it into an Ising model. For this, the following equation (116) is used.

The optimization problem solving device 100 transforms the QUABO problem into a quadratic form of spin variables, that is, an Ising model, by the equation (116). Therefore, for the Ising model formulated in this way, the optimization problem solving device 100 extracts a coefficient matrix by using a program or the like. Then, the optimization problem solving device 100 uses the extracted coefficient matrix to set the spin-to-spin coupling constant of the riser machine 10 and the local magnetic field. Then, the Ising machine 10 performs an annealing process, calculates a combination of basal spins of the Ising model (σ ₁ ^* … σ _M ^* …), and transmits the combination to the optimization problem solving device 100. The optimization problem solving device 100 receives the combination of basal spins of the Ising model calculated by the Ising machine 10.

The optimization problem solving device 100 can obtain a binary variable optimized from this combination of basis spins by using the following equation (117).

Therefore, the optimization problem solving device 100 obtains ^{a set S *} indicating the combination of the subscripts of the optimized sparse components by selecting i having the following equation (118).

By using this set S ^* , the following equation (119) can be obtained.

Then, by diagonalizing as shown in the following equation (120), the optimum compression method can be obtained as shown in the following equation (121).

The dimensionally compressed matrix C (dimensional compression matrix C) to be finally obtained is a matrix in which the ^{column components other than the S *} column components become zero vectors. Therefore, the optimization problem solving device 100 first initializes the dimensional compression matrix C to zero, and then sets _{the C S *} obtained by the equation (121) in the submatrix C (S ^* ^{) of the dimensional compression matrix C by the S * columns.} By substituting, the dimensional compression matrix C is generated. For example, the optimization problem solving device 100 substitutes _{C S *} obtained by Eq. (121) into the column corresponding to each subscript included in ^{the set S *} among the columns of the dimensional compression matrix C initialized with zero. By doing so, the dimension compression matrix C is generated.

[2-5-4. Flow example of compressed sensing optimization]
Next, the flow of compressed sensing optimization will be described with reference to FIG. FIG. 15 is a flowchart showing the procedure of the compressed sensing optimization process. FIG. 15 is an example of the flow of L0 sparse modeling by the optimization problem solving system 1 including the Ising machine 10. In the following, a case where the optimization problem solving device 100 is the main processing element is shown as an example, but the device is not limited to the optimization problem solving device 100, and may be any device included in the optimization problem solving system 1.

As shown in FIG. 15, the optimization problem solving device 100 acquires a non-sparse observation matrix (step S201). Step S201 is a process corresponding to the above formula (90) and the like.

The optimization problem solving device 100 acquires combinatorial optimization parameters for selection of observations (step S202). The optimization problem solving device 100 extracts a coefficient matrix in the formulation of combinatorial optimization. In step S202, processing from the optimization of the compressed sensing described above to the QUABO problem and the formulation into the Ising model is performed. Step S202 is a process corresponding to the above equation (104), equation (106), and the like.

The optimization problem solving device 100 combines the optimization parameters and sends them to the optimization machine (step S203). The optimization problem solving device 100 receives the optimization result from the combinatorial optimization machine (step S204). That is, the optimization problem solving system 1 obtains a combination optimized by the Ising machine 10, which is a combinatorial optimization machine. As described above, the optimization problem solving system 1 uses the Ising machine 10 to improve the search efficiency and realize a high-speed search of the compression dimension.

The optimization problem solving device 100 generates an observation selection / synthesis matrix (step S205). The optimization problem solving device 100 generates a matrix of selection dimension information and composition indicating the selection of dimensions based on the combination information of dimensions. For example, the optimization problem solving device 100 generates a set S indicating a selection dimension as information for observation selection. Further, for example, the optimization problem solving device 100 generates the dimensional compression matrix C as a composite matrix. Step S202 is a process corresponding to the above equation (98), equation (100), and the like. The optimization problem solving device 100 stores the observation selection / synthesis matrix (step S206). The optimization problem solving device 100 stores the information of the selected dimension and the synthetic matrix.

[2-5-5. Compressed sensing optimization program example]
From here, an example of a compressed sensing optimization program (function) is shown with reference to FIGS. 16 to 18. In the compressed sensing optimization program of FIGS. 16 to 18, the same points as the L0 sparse modeling program of FIGS. 10 to 12 will be omitted as appropriate. FIG. 16 is a diagram showing an example of a compressed sensing optimization processing program. Specifically, FIG. 16 shows a program PG21 which is an example of the entire program of the compressed sensing optimization process.

The program PG21 shown in FIG. 16 is mainly composed of the following blocks.
-Generate spins and describe the objective function-Extract the Ising parameter from the objective function-Send the Ising parameter to the Ising machine and receive the optimal spin arrangement-Get the dimension selection matrix from the optimal spin variable

For example, the function "sparse_observation (A, s, lambda = xxx)" on the 4th line of the program PG21 calculates the objective function of L0 sparse modeling as shown in FIG.

FIG. 17 is a diagram showing an example of a program that generates an objective function of the Ising model. FIG. 17 shows a program PG22 which is an example of a function for calculating an objective function of a QUA (Ising model). The function "sparse_observation (A, s, lambda)" shown in FIG. 17 is called by the entire program PG21 of FIG. 17 to generate an objective function (corresponding to loss in FIGS. 16 and 17).

The program PG22 shown in FIG. 17 is mainly composed of the following blocks.
-Create a binary variable that represents the selection of observation dimensions using spin variables-Create an objective function that increases as the number of dimensions used increases-Create a penalty that increases when the sparse observation matrix deviates from the identity matrix-Objective function And the penalty to get the final objective function

For example, the function "Fnorm (B)" on the second line of the program PG22 is a Frobenius norm, and calculates the number of dimensions to be used. For example, the function "matmul (A.t (), matmul (B, A))" on the third line of the program PG22 finds a penalty for sparse observation. For example, the function "UnitMatrix (M) -penalty" on the fourth line of the program PG22 finds the deviation of the observation matrix from the unit matrix. For example, the expression on line 5 of program PG22 adds a penalty to the objective function.

The parameter λ (corresponding to lambda in FIG. 17) may be set to about 0.5 to 1, or the spin (corresponding to s in FIG. 17) and the parameter λ (corresponding to lambda in FIG. 17). Alternate optimization may be performed to obtain the Lagrange undetermined multiplier method.

For example, the function "dimension_selection (A, s_optimum)" on the 7th line of the program PG1 calculates the dimension selection matrix as shown in FIG.

FIG. 18 is a diagram showing an example of a program for obtaining a measurement control matrix. Specifically, FIG. 18 shows a program PG23 which is an example of a function for obtaining a measurement control matrix from a combination variable. The function "dimension_selection (A, s_optimum)" shown in FIG. 18 is called by the entire program PG1 of FIG. 18 to generate a matrix C.

The program PG23 shown in FIG. 18 is mainly composed of the following blocks.
-Use the selected rows to select dimensions-Generate a composite matrix-Use diagonalization of the matrix, etc.

For example, the formula in the first row of the program PG23 generates a submatrix of the non-sparse observation matrix A. For example, the function "matmul (As.t (), As)" on the second line of the program PG23 generates a square matrix. For example, the function "symmeterize (AsAst)" on the third line of the program PG23 performs diagonalization. For example, "D" in the program PG23 indicates a diagonal matrix, and "P" indicates an orthogonal matrix. For example, the function "sqrt (1 / D)" on the fourth line of the program PG23 finds the square root of the inverse matrix. For example, the function "matmul (Pi, matmul (Cs, P))" on the fourth line of the program PG23 performs processing using the diagonalization matrix. For example, the function "ZeroVector ()" on the sixth line of the program PG23 generates the zero component of the dimension selection matrix. For example, the formula on the 7th line of the program PG23 is assigned to the selection dimension to generate a dimension selection matrix.

For example, the optimization problem solving device 100 stores a program (function) as shown in FIGS. 16 to 18 and a program (function) called by each program in the function information storage unit 122 (see FIG. 13), and each of them is stored. Execute the process using the program.

[3. Other Examples]
Further, the optimization problem solving system 1 described above may be used for various purposes. Hereinafter, an embodiment to which the optimization problem solving system 1 is applied will be described.

[3-1. Second Example (RFID)]
The optimization problem solving system 1 may be applied to RFID (Radio Frequency Identification) using NFC (Near Field Communication). An example of this point will be briefly described with reference to FIG. FIG. 19 is a diagram showing a second embodiment to which the optimization problem solving system is applied. The usage scene UC1 in FIG. 19 shows a usage scene when the optimization problem solving system 1 is used for RFID.

First, a brief description of RFID. RFID is a technology in which a communication device on the host side called a reader emits radio waves of a predetermined frequency to detect peripheral terminals (cards, tags). Normally, the terminal side returns a reflected wave in which unique identification information is superimposed on the radio wave from the reader, and the reader identifies peripheral terminals from the received reflected wave.

Here, consider the situation where there are multiple terminals in the vicinity. In a normal RFID, peripheral terminals in turn repeatedly establish communication with a reader and return their respective identification information according to a protocol defined in a standard. However, in the case of such time-division identification information, as the number of terminals increases, it may not be possible to detect all terminals by the set timeout time.

Therefore, for example, consider that the terminal responds to the polling from the reader at the same time (at the same clock timing) with the identification information. In this case, the identification information received by the reader from the terminals is a superposition of the reflected waves from the respective terminals. In the example of FIG. 19, a reader receives identification information from three terminals of tags # 1, # 2, and # 9 among nine terminals corresponding to each of tags # 1 to # 9.

The identification information of the tag j is represented by a D-dimensional vector like _{u j.} That is, it is assumed that the identification information is the D bit. Then, if the signal strength from the tag j received by the reader is collectively represented by D bits, it becomes _{z j} u _j _{obtained by multiplying u j} by a constant. However, z _j is a value that is not 0 if the tag j is within the detection range of the reader (the circle surrounding the tags # 1, # 2, and # 9 in FIG. 19), and is 0 if it is outside the detection range. Although u _j is a bit string to represent information, it is set as a sign string of ± 1 for convenience.

If the tags within the detection range are only a small part of all N tags, the vector z is a sparse vector. Assuming that the identification information of all N tags is registered in advance, consider a matrix U in which these are arranged in a column. That is, it is the following equation (122).

For example, if the bit length of the tag identification information is 5 and the total number of tags is 6, the matrix U becomes a matrix as shown in the following equation (123).

Using the matrix U and the vector z in equation (123), the signal strength received by the reader is represented by x = Uz. At this time, the task of simultaneous reading of RFID by the reader is formulated as the following equations (124) and (125).

Here, P in equation (125) is a set of tags whose signal strength is not 0. Since this problem is a problem of L0 sparse modeling, the following equation (126) can be formulated as an optimization represented by a combination variable.

B in equation (126) is a diagonal matrix having binary variables representing zero and non-zero of the sparse vector as diagonal components, as in the above example. The method of converting the formulation of this combinatorial optimization into a quadratic form according to the Ising model can follow the method described so far. For example, the optimization problem solving device 100 reduces the dimension B of the equation (126) and converts it into a quadratic form.

For example, ^{under the condition that the eigenvalue of the maximum absolute value of IU T} U is sufficiently smaller than 1, it can be approximated as the following equation (127).

Since the equation (127) is in the quadratic form of the binary combination variable, it is easy to embed it in the Ising model by converting the binary variable into a spin variable. The identification information represented by the formula (122) is configured by an appropriate method.

The above-mentioned compressed sensing optimization is used as the optimum selection method. Hereinafter, for the sake of simplicity, the case where the number of tags is expressed by a power of 2 and the observation matrix A without compression is used as the Hadamard matrix will be described as an example. At this time, since the row vectors are orthogonal to each other, it is formulated as the following equation (128) as the optimization of the combination variable representing the selection / non-selection of the rows of the observation matrix A.

The optimization problem solving device 100 selects the optimum row while adjusting the term for compressing as much as possible in the first term and the term for reducing the correlation between the columns in the second term as a constant given the parameter λ. This can be used to optimize the identification information.

As another optimization method, there is also a method of selecting the optimum line after keeping the number of bits constant. In this case, the objective function is changed and becomes an optimization problem as shown in the following equation (129).

Even in this case, optimization is performed within the range of the quadratic form of the binary variable, so if the binary variable is converted to a spin variable, it can be embedded in the Ising model.

[3-2. Third Example (extension of the second embodiment)]
The optimization problem solving system 1 may extend the second embodiment and be applied to spread spectrum communication, code division multiple access (CDMA), and the like. An example of this point will be briefly described below.

In the extension to spread spectrum communication, code multiplex connection, etc., the tag identification information u _j is considered to be a spread code, and the tag subscript j is considered to be one for the multiplexed channel.

The sign of the j-th component z _j of the sparse solution z ^* changes periodically according to the sign of the information being multiplexed. Therefore, _{by reading the code of z j} , the information transmitted on channel j can be read. This spread spectrum can be used to receive information from the terminal at the same time on the base station side at the time of uplink.

On the base station side, assuming that all diffusion codes are known and that few channels are used at the same time, sparse modeling is used ^{to obtain the sparse solution z *,} and the information transmitted for each channel from the code of each component is obtained. read. As for the optimization of the diffusion code, the same method as the RFID identification information shown in the second embodiment may be used, and thus the description thereof will be omitted.

[3-3. Fourth Example (Target Detection)]
The optimization problem solving system 1 may be applied to the detection of a target by an array radar or a rider. For example, an optimization problem solving system 1 may be used for detecting an obstacle in front of a vehicle with an in-vehicle radar or rider. An example of this point will be briefly described with reference to FIGS. 20 to 22. 20 to 22 are diagrams showing a fourth embodiment to which the optimization problem solving system is applied.

The usage scene UC2 in FIG. 20 shows a usage scene when the optimization problem solving system 1 is used for target detection. Specifically, the usage scene UC2 in FIG. 20 is an image diagram of an obstacle in the direction of the azimuth angle -90 degrees to 90 degrees in front of the vehicle, and when there is an obstacle having various depths and moving speeds. Is shown as an example.

When the incoming wave from an obstacle (target) (the incoming wave reflected by the radar) is measured using a radar, the distance, angle, and velocity component to the obstacle (target) can be known. Of these, if only the incoming wave whose distance and speed are within a predetermined range is selected in advance, only the target moving at the same distance and the same speed is selected. Therefore, the target position may be obtained by obtaining the angle component of the incoming wave. ..

The estimation of the target azimuth angle from the received signal of the array radar by sparse modeling is disclosed in the following documents, for example.
・ A Sparse Signal Reconstruction Perspective for Source Localization With Sensor Arrays, Dmitry Malioutov et al. <Http://ssg.mit.edu/~willsky/publ_pdfs/175_pub_IEEE.pdf>

When using sparse modeling according to the method in the above document, the angle of the incoming wave is discretized from -90 degrees to 90 degrees in front to a grid of, for example, in 1 degree increments. Then, the usage scene UC21 in FIG. 21 becomes a sparse vector (a non-zero component where there is a target and zero where there is no target).

The information of the incoming wave generated from this sparse vector is received by the array radar having multiple antennas. At this time, if the observation matrix (direction matrix) that generates the received signal of the incoming wave received by a plurality of antennas is known, the target can be detected at an azimuth angle having a resolution higher than the number of antennas by using sparse modeling.

In this way, using sparse modeling to detect a target at a higher resolution grid azimuth with a relatively small number of antennas is the super-resolution of the azimuth of the incoming wave in the radar.

The usage scene UC23 in FIG. 22 shows the above contents in a two-dimensional image. Regarding the usage scene UC23 of FIG. 22, in order to handle it in sparse modeling, an observation matrix that generates a received signal of the radar array and received signal information of the incoming wave are required.

In the case of an array radar, as shown in the following equations (130) and (131), the observation matrix U is expressed as a matrix in which the steering vectors u (θ _k _{) with respect to the arrival angle θ k are arranged in the column direction.}

Here, M in the equation (130) corresponds to the number of discrete grids of angles, and is 180 in FIG. 22. Further, N in the equation (131) is the number of receiving antennas of the array radar, for example, 10. _{The d k} of the equation (131) is the distance from the rightmost antenna to the kth antenna in the figure, and is often set as the following equation (132) in order to reduce aliasing. ..

At this time, the received signal x of the incoming wave is modeled by the following equation (133) using the vector z of the zero-non-zero component for each grid and the observation matrix A.

Then, the problem of each estimation of the arrival of the radar by sparse modeling (L1 sparse modeling) is formulated as the following equation (134).

In this way, when the conventional method receives the received signal of the incoming wave once and estimates the azimuth angle of the target from it, normal L1 sparse modeling (also called Lasso) has been used.

On the other hand, in the present disclosure, the condition for L1 sparse modeling to reach the true solution is narrower than the condition for L0 sparse modeling to reach the true solution. L0 sparse modeling is adopted.

Whereas the conventional method is _{a problem of minimizing the L1 norm || z || 1} , in the fourth embodiment, it is a problem of minimizing the _{L0 norm || z || 0 shown in.} There is.

In the above L0 sparse modeling and L1 sparse modeling, the observation matrix, the received signal, and the sparse solution are complex numbers. In complex space, use Hermitian conjugates instead of transpose matrices and vectors. As a result, the optimization problem solving device 100 performs the formulation of the following equation (136) (the transpose symbol T is the symbol of Elmeat conjugate †) as an optimization that expresses L0 sparse modeling as a combination variable. be able to.

B in equation (136) is a diagonal matrix having binary variables representing zero and non-zero of the sparse vector as diagonal components, as in the above example.

The method of converting this combinatorial optimization formulation into a quadratic form according to the Ising model can follow the method described so far. For example, the optimization problem solving device 100 reduces the dimension B of the equation (136) and converts it into a quadratic form.

For example, ^{under the condition that the eigenvalue of the maximum absolute value of IA †} A is sufficiently smaller than 1, it can be approximated by the following equation (137).

Since the equation (137) is in the quadratic form of the binary combination variable, it is easy to embed it in the Ising model by converting the binary variable into a spin variable.

Note that the radar super-resolution in the above document also considers super-resolution with multiple snapshots.

In this case, the received signal is the following equation (139) using the matrices X and Z in which the single received signal x and its solution z shown in the following equation (138) are arranged in the column direction by the number of snapshots. It is represented by.

In the above document, the received signal for the number of snapshots is actually decomposed into singular values, and the received signal component using the basis vector corresponding to the singular value having a large absolute value is modeled. Specifically, the singular value is decomposed as in the following equation (140), and K bases are selected in order from the one having the largest absolute value of the singular value to obtain the following equation (141).

_{For Z as well, let K be a matrix Z SV in} which K are arranged in the column direction instead of the number of snapshots. Then, the problem of L1 sparse modeling is expressed by the group Lasso as in the following equation (142).

However, the following number (143) is the L2 norm of the vector with i rows of _{z SV.}

When the equation (142) is expressed by L0 sparse modeling, it can be expressed by the following equation (144).

L0 sparse modeling that takes into account the elements of such a group is expressed by the following equation (145).

X in the equation (145) may be an arrangement of snapshots as they are, or may be _{an X SV in which K pieces are selected in order from the one having the largest absolute value of the singular value.}

Further, ^{under the condition that the eigenvalue of the maximum absolute value of IA †} A is sufficiently smaller than 1, it can be approximated by the following equation (146).

In this approximation, the objective function is in the quadratic form of the binary variable, and it can be embedded in the rising model by converting the binary variable into a spin variable. Thus, in the fourth embodiment, it is possible to extend not only to real space but also to complex space, and it is possible to extend so as to impose group constraints on the solution. The number of array radars may be determined (set) by an appropriate method. The method for optimally selecting an array radar utilizes the compressed sensing optimization described above.

[3-4. Fifth Example (Image Diagnosis)]
The optimization problem solving system 1 may be applied to the field of diagnostic imaging such as MRI. This point will be briefly described with reference to FIG. 23, using MRI as an example. The optimization problem solving system 1 is not limited to MRI, and may be applied to various diagnostic imaging. FIG. 23 is a diagram showing a fifth embodiment to which the optimization problem solving system is applied. The optimization problem solving system 1 of FIG. 23 includes an MRI apparatus 500.

As shown in FIG. 23, in the optimization problem solving system 1, it can be used for compressed sensing by MRI or the like. The following documents are disclosed regarding compressed sensing of MRI, and detailed description thereof will be omitted.
・ Sparse MRI: The application of compressed sensing for rapid MR imaging, Michael Lustig et al. <Https://onlinelibrary.wiley.com/doi/epdf/10.1002/mrm.21391>

The MRI apparatus 500 is an diagnostic imaging apparatus that includes a control computer 510, a radio wave transmission / reception device 520, and a detector unit 530, and uses a nuclear magnetic resonance phenomenon to image information inside the living body. .. For example, the control computer 510 controls each configuration of the radio wave transmission / reception device 520 and the detector unit 530. For example, the radio wave transmitter / receiver 520 is a radio frequency transmitter / receiver that performs sensing. The detector unit 530 has a configuration such as a magnet, a detection coil, and a magnetic field gradient coil. The MRI apparatus 500 may have a display unit (display device) such as a display and an input unit (input device) such as a mouse or a keyboard. Since the MRI apparatus 500 is a so-called magnetic resonance imaging (MRI), detailed description thereof will be omitted.

From here, I will explain in which block L0 sparse modeling and compression optimization are performed in the overall picture of MRI. First, in the optimization problem solving system 1, compression optimization is used to thin out observations. In the optimization problem solving system 1 of FIG. 23, the control computer 510 of the MRI device 500 functions as the measurement controller shown in the block B6 of the optimization problem solving system 1, and the radio wave transmission / reception device 520 of the MRI device 500 is in the block B7. Functions as an indicator sensor. In the fifth embodiment, the sensor 50 does not have to be included in the optimization problem solving system.

The control computer 510, which functions as a measurement controller, realizes the measurement while thinning out the measurement according to the information of the observation dimension selected so that the column vectors of the observation matrix are as uncorrelated as possible by using the combinatorial optimization machine in advance.

Further, one of a plurality of Ising machines 10 is used as the combinatorial optimization machine, and the selection criterion is that the processing is offline, so that the accuracy is better than the high speed and the real-time property. It becomes a standard.

On the other hand, the data restoration of compressed sensing using L1 sparse modeling is usually used for the processing after block B7, but here, the data restoration from the sparse solution of L0 sparse modeling is performed.

This L0 sparse modeling is also realized by using a combinatorial optimization machine such as the Ising machine 10. The standard at this time is different from the case of optimization of the compression dimension, and since this process is online, real-time performance is emphasized. This is because we expect that data will be restored with low delay each time information from the sensor is input one after another.

[3-5. Sixth Example (Other)]
The application destination of the optimization problem solving system 1 is not limited to the above, and may be applied to various applications in which sparse modeling is used. This point will be briefly described below.

For example, in the fourth embodiment, obstacle detection by an in-vehicle radar is shown as an example, but a rider may be used instead of the radar. In addition to obstacle detection, radar may be used for weather condition observation, underground and water observation, and remote sensing from aircraft (drones) and satellites.

In addition, as an application using radar, for example, super-resolution imaging of black holes is known in radio astronomical observation by radio observatory, but sparse modeling used in this case is realized by quantum annealing and Ising machine. Is also good.

As an application related to radar azimuth estimation, sparse modeling of sound source direction estimation can also be considered.

Examples of RFID and spread spectrum communication are examples of sparse modeling utilization in communication, but in the communication field, we also used wireless multipath estimation, effective bandwidth utilization by detecting the usage status of multiple bands, and network tomography. It can be applied to the estimation of network delay points and wireless sensor nets.

Furthermore, as an example similar to the example of shortening the MRI time, it is conceivable to reduce the power consumption of the imaging sensor. In this case, processing for optimizing the element to be sensed in advance and restoration of the actual sensing result by sparse modeling can be considered.

As described above, the optimization problem solving system 1 of the present disclosure can be applied to various sparse modeling applications already known.

In the above world, the optimization problem solving system 1 converts L1 sparse modeling used in various compressed sensing and super-resolution applications into L0 sparse modeling, which is a more ideal sparse modeling. When the optimization problem solving system 1 replaces L1 sparse modeling with L0 sparse modeling, it connects to a combinatorial optimization dedicated computer such as a singing machine, and in order to connect to them, the problem of L0 sparse modeling is combined in a quadratic form. Perform a formulation to convert to optimization.

[4. Other configuration examples, etc.]
The processing according to the above-described embodiment or modification may be carried out in various different forms (modifications) other than the above-mentioned embodiment or modification.

[4-1. Other configuration examples]
In the above example, the case where the optimization problem solving device 100 and the rising machine 10 are separate bodies is shown, but the optimization problem solving device 100 and the rising machine 10 may be integrated. For example, when the Ising machine 10 is realized by a digital circuit without using superconductivity, the Ising machine 10 may be arranged on the edge side. For example, when the calculation using the Ising model is performed on the edge side, the optimization problem solving device 100 and the Ising machine 10 may be integrated.

[4-2. Information generation method used for processing programs and parameters]
A method for generating the optimization problem solving process and parameters described above may be provided. Further, a method of generating a program used by the Ising machine 10 described above to execute a calculation may be provided.

[4-3. others]
Further, among the processes described in each of the above embodiments, all or part of the processes described as being automatically performed can be manually performed, or the processes described as being manually performed. It is also possible to automatically perform all or part of the above by a known method. In addition, information including processing procedures, specific names, various data and parameters shown in the above documents and drawings can be arbitrarily changed unless otherwise specified. For example, the various information shown in each figure is not limited to the information shown in the figure.

Further, each component of each device shown in the figure is a functional concept, and does not necessarily have to be physically configured as shown in the figure. That is, the specific form of distribution / integration of each device is not limited to the one shown in the figure, and all or part of them may be functionally or physically distributed / physically in any unit according to various loads and usage conditions. Can be integrated and configured.

Further, each of the above-described embodiments and modifications can be appropriately combined as long as the processing contents do not contradict each other.

Further, the effects described in the present specification are merely examples and are not limited, and other effects may be obtained.

[5. Effect of this disclosure]
As described above, in the optimization problem solving method according to the present disclosure, the first data from the sensor (sensor 50 in the embodiment) is input, and an objective function for sparse modeling the first data is generated. A coefficient matrix related to a variable to be optimized in the objective function is generated, the coefficient matrix is transmitted to the first Ising machine that performs combinatorial optimization calculation, and the coefficient matrix is received from the first Ising machine (Ising machine 10 in the embodiment). Generate the optimum solution of sparse modeling based on the data of 2.

In this way, the optimization problem solving method generates a coefficient matrix corresponding to the objective function derived by sparse modeling for the input of the first data, sends it to the rising machine, and receives it from the rising machine. Generate an optimal solution based on the data in. As a result, the optimization problem solving method can appropriately generate a solution of the optimization problem for the data from the sensor by using the Ising machine. That is, the computer that executes the optimization problem solving method can appropriately generate the solution of the optimization problem by using the singing machine for the data from the sensor.

Also, sparse modeling is L0 sparse modeling. In this way, the optimization problem solving method can generate the optimum solution by L0 sparse modeling, and can appropriately generate the solution of the optimization problem using the Ising machine for the data from the sensor. ..

The variable to be optimized is a binary variable that distinguishes between a non-zero component and a zero component for each component of the sparse variable that models the first data. Thus, the optimization problem solving method can generate a solution that optimizes the binary variable that distinguishes between the non-zero component and the zero component for each component of the sparse variable that models the first data. For the data from the sensor, the solution of the optimization problem can be appropriately generated by using the singing machine.

The variable to be optimized is a bit variable obtained by quantizing each component of the sparse variable that models the first data. In this way, it is possible to generate a solution that optimizes each bit variable obtained by quantization for each component of the sparse variable that models the first data, and the Ising machine for the data from the sensor. Can be used to properly generate solutions to optimization problems.

Also, generate a coefficient matrix corresponding to the objective function converted to the QUAB format. In this way, the optimization problem solving method is to send the coefficient matrix corresponding to the objective function converted to the QUAB format to the Ising machine, and then to the data from the sensor, the optimization problem using the Ising machine. Can be properly generated.

The coefficient matrix is an array composed of coefficients related to the first-order or higher terms of the variables to be optimized extracted from the objective function. As a result, the optimization problem solving method is to send a coefficient matrix, which is an array composed of coefficients related to the first-order or higher terms of the variable to be optimized, extracted from the objective function to the Ising machine, from the sensor. The solution of the optimization problem can be appropriately generated for the data by using the singing machine.

The objective function is a function corresponding to the first data. In this way, the optimization problem solving method uses an objective function corresponding to the first data to generate an optimum solution, thereby solving the optimization problem for the data from the sensor using a singing machine. Can be generated appropriately.

Also, the optimal solution is the solution corresponding to the first data. In this way, the optimization problem solving method can appropriately generate the solution of the optimization problem by using the Ising machine for the data from the sensor by generating the solution corresponding to the first data. can.

In addition, the optimum solution is calculated by calculating the value of the non-zero component using the second data. In this way, the optimization problem solving method uses a singing machine for the data from the sensor by calculating the optimum solution by calculating the value of the non-zero component using the second data. The solution of the optimization problem can be generated appropriately.

The first Ising machine is a quantum computer (Ising machine 10a in the embodiment) or a quantum-conceived computer (Ising machine 10b in the embodiment). In this way, the optimization problem solving method is to send a coefficient matrix to a quantum computer or a quantum-conceived computer and generate an optimum solution based on the second data received from the quantum computer or the quantum-conceived computer. The solution of the optimization problem can be appropriately generated by using the singing machine for the data from.

Further, in the optimization problem solving method according to the present disclosure, the dimension selection of the first data is further performed by the measurement controller. As a result, the optimization problem solving method can generate the optimum solution for the data whose dimension is selected, and appropriately generate the solution of the optimization problem for the data from the sensor using the singing machine. can do.

In addition, the optimization problem solving method according to the present disclosure uses a sparse observation model for dimension selection. As a result, the optimization problem solving method can generate the optimum solution for the data whose dimensions are selected using the sparse observation model, and the optimization problem for the data from the sensor using the singing machine. Can be properly generated.

Further, the sparse observation model is generated by a second Ising machine (Ising machine 10 in the embodiment) that performs observation dimension selection modeling. In this way, the optimization problem solving method can generate the optimum solution for the data whose dimensions are selected using the sparse observation model generated by the singing machine, and the singing for the data from the sensor. Machines can be used to properly generate solutions to optimization problems.

Further, in the optimization problem solving method according to the present disclosure, the user further selects the first rising machine by the user interface (in the embodiment, the input unit 140 and the display unit 150 of the optimization problem solving device 100). Thereby, the optimization problem solving method can appropriately generate a solution of the optimization problem by using the Ising machine desired by the user.

Further, in the optimization problem solving method according to the present disclosure, the user interface presents a plurality of selection candidates to the user, and among the plurality of selection candidates, a coefficient is applied to the first rising machine corresponding to the selection candidate selected by the user. Send the matrix. As a result, the optimization problem solving method can appropriately generate a solution of the optimization problem using the Ising machine desired by the user by performing processing using the Ising machine selected by the user from a plurality of candidates. can.

Further, in the optimization problem solving method according to the present disclosure, the user interface presents detailed information of the selection candidate specified by the user among the plurality of selection candidates. As a result, in the optimization problem solving method, the user can confirm the details of each candidate and select the Ising machine to be used, so that the user's satisfaction can be improved.

Further, in the optimization problem solving method according to the present disclosure, the detailed information includes at least one of the usage fee, specifications, and recommended use of the first Ising machine corresponding to the selection candidate. As a result, in the optimization problem solving method, the user can select the Ising machine to be used by confirming the usage fee, specifications, recommended usage, etc. of each candidate, so that the user's satisfaction can be enhanced.

As described above, the optimization problem solving device (optimization problem solving device 100 in the embodiment) according to the present disclosure includes a sensor input unit (reception unit 131 in the embodiment) for inputting the first data from the sensor. A first generation unit (first generation unit 132 in the embodiment) that generates an objective function for sparse modeling of the first data, and a second generation unit (first generation unit 132) that generates a coefficient matrix related to a variable to be optimized in the objective function. In the embodiment, the second generation unit 133), the transmission unit that transmits the coefficient matrix to the first Ising machine that performs the combinatorial optimization calculation (the transmission unit 134 in the embodiment), and the second receiving unit received from the first Ising machine. A third generation unit (third generation unit 135 in the embodiment) that generates an optimum solution for sparse modeling based on the above data is provided.

In this way, the optimization problem solving device generates a coefficient matrix corresponding to the objective function derived by sparse modeling for the input of the first data, sends it to the rising machine, and receives it from the rising machine. Generate an optimal solution based on the data in. As a result, the optimization problem solving method can appropriately generate a solution of the optimization problem for the data from the sensor by using the Ising machine.

[6. Hardware configuration]
The information device such as the optimization problem solving device 100 according to each of the above-described embodiments and modifications is realized by a computer 1000 having a configuration as shown in FIG. 24, for example. FIG. 24 is a hardware configuration diagram showing an example of a computer that realizes the function of the optimization problem solving device. Hereinafter, the optimization problem solving device 100 will be described as an example. The computer 1000 has a CPU 1100, a RAM 1200, a ROM (Read Only Memory) 1300, an HDD (Hard Disk Drive) 1400, a communication interface 1500, and an input / output interface 1600. Each part of the computer 1000 is connected by a bus 1050.

The CPU 1100 operates based on the program stored in the ROM 1300 or the HDD 1400, and controls each part. For example, the CPU 1100 expands the program stored in the ROM 1300 or the HDD 1400 into the RAM 1200, and executes processing corresponding to various programs.

The ROM 1300 stores a boot program such as a BIOS (Basic Input Output System) executed by the CPU 1100 when the computer 1000 is started, a program depending on the hardware of the computer 1000, and the like.

The HDD 1400 is a computer-readable recording medium that non-temporarily records a program executed by the CPU 1100 and data used by such a program. Specifically, the HDD 1400 is a recording medium for recording an information processing program such as an optimization problem solving processing program according to the present disclosure, which is an example of program data 1450.

The communication interface 1500 is an interface for the computer 1000 to connect to an external network 1550 (for example, the Internet). For example, the CPU 1100 receives data from another device or transmits data generated by the CPU 1100 to another device via the communication interface 1500.

The input / output interface 1600 is an interface for connecting the input / output device 1650 and the computer 1000. For example, the CPU 1100 receives data from an input device such as a keyboard or mouse via the input / output interface 1600. Further, the CPU 1100 transmits data to an output device such as a display, a speaker, or a printer via the input / output interface 1600. Further, the input / output interface 1600 may function as a media interface for reading a program or the like recorded on a predetermined recording medium (media). The media is, for example, an optical recording medium such as DVD (Digital Versatile Disc) or PD (Phase change rewritable Disk), a magneto-optical recording medium such as MO (Magneto-Optical disk), a tape medium, a magnetic recording medium, or a semiconductor memory. Is.

For example, when the computer 1000 functions as the optimization problem solving device 100, the CPU 1100 of the computer 1000 may execute an information processing program such as an optimization problem solving processing program loaded on the RAM 1200 to control the control unit 130 or the like. Realize the function. Further, the HDD 1400 stores an information processing program such as an optimization problem solving processing program according to the present disclosure, and data in the storage unit 120. The CPU 1100 reads the program data 1450 from the HDD 1400 and executes the program, but as another example, these programs may be acquired from another device via the external network 1550.

The present technology can also have the following configurations.
(1)
Enter the first data from the sensor,
Generate an objective function for sparse modeling of the first data,
Generate a coefficient matrix for the variables to be optimized in the objective function,
The coefficient matrix is transmitted to the first Ising machine that performs the combinatorial optimization calculation.
An optimization problem solving method that generates an optimum solution for sparse modeling based on the second data received from the first Ising machine.
(2)
The sparse modeling is the L0 sparse modeling. The optimization problem solving method according to (1).
(3)
The variable to be optimized is a binary variable that distinguishes between a non-zero component and a zero component for each component of the sparse variable that models the first data. The optimization problem according to (1) or (2). Solution.
(4)
The variable to be optimized is the optimization described in any one of (1) to (3), which is a bit variable obtained by quantizing each component of the sparse variable that models the first data. How to solve the problem.
(5)
The optimization problem solving method according to any one of (1) to (4), which generates the coefficient matrix corresponding to the objective function converted into the QUAB format.
(6)
The optimization problem solving method according to (5), wherein the coefficient matrix is an array composed of coefficients related to first-order or higher terms of variables to be optimized extracted from the objective function.
(7)
The optimization problem solving method according to any one of (1) to (6), wherein the objective function is a function corresponding to the first data.
(8)
The optimization problem solving method according to any one of (1) to (7), wherein the optimum solution is a solution corresponding to the first data.
(9)
The optimization problem solving method according to (8), wherein the optimum solution is calculated by calculating the value of the non-zero component using the second data.
(10)
The optimization problem solving method according to any one of (1) to (9), wherein the first isging machine is a quantum computer or a quantum-conceived computer.
(11)
Further, the dimension selection of the first data is performed by the measurement controller.
The optimization problem solving method according to any one of (1) to (10).
(12)
The dimension selection uses a sparse observation model.
The optimization problem solving method according to (11).
(13)
The sparse observation model is generated by a second Ising machine that performs observation dimension selection modeling.
The optimization problem solving method described in (12).
(14)
Further, the user interface allows the user to select the first Ising machine.
The optimization problem solving method according to any one of (1) to (13).
(15)
The user interface presents the user with a plurality of selection candidates.
The optimization problem solving method according to (14), wherein the coefficient matrix is transmitted to the first Ising machine corresponding to the selection candidate selected by the user among the plurality of selection candidates.
(16)
The optimization problem solving method according to (15), wherein the user interface presents detailed information of a selection candidate designated by the user among the plurality of selection candidates.
(17)
The optimization problem solving method according to (16), wherein the detailed information includes at least one of the usage fee, specifications, and recommended applications of the first Ising machine corresponding to the selection candidate.
(18)
A sensor input unit that inputs the first data from the sensor,
A first generator that generates an objective function for sparse modeling the first data,
The second generator that generates the coefficient matrix for the variable to be optimized in the objective function,
A transmitter that transmits the coefficient matrix to the first Ising machine that performs combinatorial optimization calculations,
A third generation unit that generates an optimum solution for sparse modeling based on the second data received from the first Ising machine.
Optimized problem solving device.

1 Optimization problem solving system 100 Optimization problem solving device 110 Communication unit 120 Storage unit 121 Data storage unit 122 Function information storage unit 130 Control unit 131 Reception unit 132 1st generation unit 133 2nd generation unit 134 Transmission unit 135 3rd generation Unit 140 Input unit 150 Display unit 10 Rising machine 11 Communication unit 12 Storage unit 13 Quantum device unit 14 Control unit 141 Acquisition unit 142 Calculation unit 143 Transmission unit 50 Sensor

Claims

Enter the first data from the sensor,
Generate an objective function for sparse modeling of the first data,
Generate a coefficient matrix for the variables to be optimized in the objective function,
The coefficient matrix is transmitted to the first Ising machine that performs the combinatorial optimization calculation.
An optimization problem solving method that generates an optimum solution for sparse modeling based on the second data received from the first Ising machine.
The optimization problem solving method according to claim 1, wherein the sparse modeling is L0 sparse modeling.
The optimization problem solving method according to claim 1, wherein the variable to be optimized is a binary variable that distinguishes between a non-zero component and a zero component for each component of the sparse variable that models the first data.
The optimization problem solving method according to claim 1, wherein the variable to be optimized is a bit variable obtained by quantizing each component of the sparse variable that models the first data.
The optimization problem solving method according to claim 1, wherein the coefficient matrix corresponding to the objective function converted into a QUADOTIC Unconstrained Binary Optimization (QUADRO) format is generated.
The optimization problem solving method according to claim 5, wherein the coefficient matrix is an array composed of coefficients related to first-order or higher terms of variables to be optimized extracted from the objective function.
The optimization problem solving method according to claim 1, wherein the objective function is a function corresponding to the first data.
The optimization problem solving method according to claim 1, wherein the optimum solution is a solution corresponding to the first data.
The optimization problem solving method according to claim 8, wherein the optimum solution is calculated by calculating the value of the non-zero component using the second data.
The optimization problem solving method according to claim 1, wherein the first isging machine is a quantum computer or a quantum-conceived computer.
Further, the dimension selection of the first data is performed by the measurement controller.
The optimization problem solving method according to claim 1.
The dimension selection uses a sparse observation model.
The optimization problem solving method according to claim 11.
The sparse observation model is generated by a second Ising machine that performs observation dimension selection modeling.
The optimization problem solving method according to claim 12.
Further, the user interface allows the user to select the first Ising machine.
The optimization problem solving method according to claim 1.
The user interface presents the user with a plurality of selection candidates.
The optimization problem solving method according to claim 14, wherein the coefficient matrix is transmitted to the first Ising machine corresponding to the selection candidate selected by the user among the plurality of selection candidates.
The optimization problem solving method according to claim 15, wherein the user interface presents detailed information of the selection candidate designated by the user among the plurality of selection candidates.
The optimization problem solving method according to claim 16, wherein the detailed information includes at least one of a usage fee, a specification, and a recommended application of the first Ising machine corresponding to a selection candidate.
A sensor input unit that inputs the first data from the sensor,
A first generator that generates an objective function for sparse modeling the first data,
The second generator that generates the coefficient matrix for the variable to be optimized in the objective function,
A transmitter that transmits the coefficient matrix to the first Ising machine that performs combinatorial optimization calculations,
A third generation unit that generates an optimum solution for sparse modeling based on the second data received from the first Ising machine.
Optimized problem solving device.